# Multivariable Calculus

## Vector Calculus

### Vector Differentiation

Solution If $\displaystyle A=xyz \vec i+xz^2 \vec j-y^3 \vec k, B=x^3 \vec i-xyz \vec j+x^2 z \vec k \,$ calculate $\displaystyle \frac {\partial ^2 \vec A}{\partial y^2}\times \frac{\partial ^2 \vec B}{\partial x^2}$ at the point (1,1,0).
Solution Find $\displaystyle \frac{dr}{dt},\frac{d^2 r}{dt^2}$ when $\displaystyle r=3i-6t^2 j+4t k$

Solution If $\displaystyle r=\sin t i + \cos t j + t k \,$ Find $\displaystyle \frac{dr}{dt},\frac{d^2 r}{dt^2},\left |\frac{dr}{dt}\right\vert,\left |\frac{d^2 r}{dt^2}\right\vert$

Solution If $\displaystyle r = \cos nt i + \sin nt j\,$ where n is constant and t varies, prove that $\displaystyle r\times (\frac{dr}{dt})=nk\,$ and $\displaystyle r \cdot (\frac{dr}{dt})=0\,$ .

Solution If $\displaystyle r=e^ {nt} a + e^ {-nt} b \,$ where a,b are constant vectors, show that $\displaystyle (\frac{d^2 r}{dt^2})-n^2 r =0\,$

Solution If $\displaystyle r=a\cos \omega t + b\sin \omega t \,$ ,Show that $\displaystyle r\times \frac{dr}{dt}=\omega a \times b\,$ and $\displaystyle \frac{d^2 r}{dt^2}= - \omega^2 r\,$

Solution If $\displaystyle u=t^2 i -tj+(2t+1)k\,$ and $\displaystyle v=(2t-3)i+j-t k\,$ , Find $\displaystyle \frac{d}{dt} (u\cdot v)\,$ and $\displaystyle \frac{d}{dt} (u\times v)\,$ where t=1.

Solution If $\displaystyle a=\sin \theta i+\cos \theta j+\theta k,b=\cos \theta i-\sin \theta j-3k,c=2i+3j-k\,$ .Find $\displaystyle \frac{d}{d\theta}[a\times(b\times c)]\,$ at $\displaystyle \theta=0\,$

Solution A particle moves along a curve whose parametric equations are $\displaystyle x=e^{-t},y=2\cos 3t,z=\sin 3t\,$ .Find the velocity and acceleration at t=0.

Solution A particle moves along the curve $\displaystyle x=t^3+1,y=t^2,z=2t+5\,$ where t is the time.Find the components of its velocity and acceleration at t=1 in the direction of $\displaystyle i+j+3k\,$ .

Solution A particle moves so that its position vector is given by $\displaystyle r=\cos \omega t i+\sin \omega t j\,$ where $\displaystyle \omega\,$ is a constant.Show that i).The velocity of the particle is perpendicular to r ii).The acceleration is directed towards the origin and has magnitude proportional to the distance from the origin.

Solution Show that if a,b,c are constant vectors,then $\displaystyle r=at^2+bt+c\,$ is the path of a particle moving with constant acceleration.

SolutionIf $\displaystyle f=\cos xy i+(3xy-2x^2)j-(3x+2y)k\,$ ,find the value of $\displaystyle \frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial ^2 f}{\partial x^2},\frac{\partial ^2 f}{\partial y^2},\frac{\partial ^2 f}{\partial x \partial y}\,$ .

Solution If $\displaystyle f=(2x^2 y-x^4)i+(e^{xy}-y\sin x)j+x^2 \cos y k\,$ .Verify that $\displaystyle \frac{\partial ^2 f}{\partial x \partial y}=\frac{\partial ^2 f}{\partial y \partial x}\,$ .

Solution If $\displaystyle \phi(xyz)=xy^2 z\,$ and $\displaystyle f=xzi-xyj+yz^2 k\,$ .Find $\displaystyle \frac{\partial ^3 (\phi f)}{\partial x^2 \partial z}\,$ at (2,-1,1).

### Vector Integration

Solution If $\displaystyle f(t)=(t-t^2)i+2t^3j-3k\,$ ,Find i).$\displaystyle \int f(t)\,dt\,$ ii).$\displaystyle \int_1^2 f(t)\,dt\,$

Solution If $\displaystyle f(t)=ti+(t^2-2t)j+(3t^2+3t^3)k\,$ ,find $\displaystyle \int_0^1 f(t)\,dt\,$ .

Solution Evaluate $\displaystyle \int_0^1 (e^t i+e^{-2t} j+tk)\,dt\,$

Solution If $\displaystyle r=ti-t^2j+(t-1)k\,$ and $\displaystyle s=2t^2i+6tk\,$ ,Evaluate $\displaystyle \int_0^2 r\cdot s \,dt \,$ and $\displaystyle \int_0^2 r\times s\, dt \,$ .

Solution Evaluate $\displaystyle \int_0^2 a\cdot b\times c \,dt\,$ where $\displaystyle a=ti-3j+2tk,b=i-2j+2k,c=3i+tj-k\,$

Solution Given that $\displaystyle r(t)=2i-j+2k\,$ when t=2,$\displaystyle r(t)=4i-2j+3k\,$ when t=3, Show that $\displaystyle \int_2^3 [r\cdot \frac{dr}{dt}]\,dt = 10\,$ .

Solution Evaluate $\displaystyle \int_1^2 r\times \frac{d^2 r}{dt^2}\,dt\,$ where $\displaystyle r=2t^2i+tj-3t^3k\,$

Solution If $\displaystyle r(t)=5t^2i+tj-r^3k\,$ ,prove that $\displaystyle \int_1^2 r\times \frac{d^2 r}{dt^2}\,dt=-14i+75j-15k\,$

Solution Evaluate $\displaystyle \int a\cdot [r\times \frac{d^2 r}{dt^2}]\,dt\,$

Solution Evaluate $\displaystyle \int_1^2 [a\cdot (b\times c)+a\times(b\times c)]\,dt\,$ where $\displaystyle a=ti-3j+2tk,b=i-2j+2k,c=3i+tj-k\,$

Solution The acceleration of a moving particle at any time t is given by $\displaystyle \frac{d^2 r}{dt^2}=12\cos 2t i-8\sin 2t j+16t k\,$ .Find the velocity v and displacement r at any time t,if t=0,v=0 and r=0.

Solution Find the value of r satisfying the equation $\displaystyle \frac{d^2 r}{dt^2}=6ti-24t^2j+4\sin t k\,$ given that $\displaystyle r=2i+j,\frac{dr}{dt}=-i-3k\,$ at t=0.

Solution If the acceleration of a particle at any time t greater than or equal to zero is given by $\displaystyle a=3\cos t i+4\sin t j+t^2 k\,$ and the velocity v and displacement r are zero at t=0, then find v and r at any time t.

Solution Integrate $\displaystyle \frac{d^2 r}{dt^2}=-n^2 r\,$

SolutionIf $\displaystyle f(x,y,z)=x^3+y^3+z^3+3xyz\,$ then find $\displaystyle \nabla f \,$

SolutionIf $\displaystyle f(x,y,z)=3x^2y-y^3z^2\,$ ,find $\displaystyle \nabla f\,$ at the point (1,-2,-1).

Solution If $\displaystyle r=xi+yj+zk\,$ and $\displaystyle r=|r|=(x^2+y^2+z^2)^{\frac{1}{2}}\,$ , Prove that

i). $\displaystyle \nabla f(r)=f'(r)\nabla f\,$ ii). $\displaystyle \nabla r=(\frac{1}{r})r\,$

Solution If $\displaystyle \phi (x,y,z)=(3r^2-4r^{\frac{1}{2}}+6r^{-\frac{1}{3}})\,$ ,Show that $\displaystyle \nabla \phi=2(3-r^{-\frac{3}{2}}-r^{-\frac{7}{3}})r\,$

Solution If $\displaystyle u=x+y+z,v=x^2+y^2+z^2,w=xy+yz+zx\,$ Prove that $\displaystyle \mathrm{grad}u\cdot [\nabla v\times \nabla w]=0\,$

Solution Evaluate $\displaystyle \nabla e^{r^2}\,$ where $\displaystyle r^2=x^2+y^2+z^2\,$

Solution Show that $\displaystyle (a\cdot \nabla)\phi=a\cdot \nabla \phi\,$

Solution If $\displaystyle F=[y\frac{\partial f}{\partial z}-z\frac{\partial f}{\partial y}]i+[z\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial z}]j+[x\frac{\partial f}{\partial y}-y\frac{\partial f}{\partial x}]k\,$ ,Prove that i).$\displaystyle F=r\times \nabla f\,$ ii).$\displaystyle F\cdot r=0\,$ iii).$\displaystyle F\cdot \nabla f=0\,$

SolutionIf $\displaystyle u=3x^2 y,v=xz^2-2y\,$ ,find $\displaystyle (\nabla u)\cdot (\nabla v)\,$

SolutionFind $\displaystyle \nabla \phi,|\nabla \phi|\,$ where $\displaystyle \phi (x,y,z)=(x^2+y^2+z^2)e^{-(x^2+y^2+z^2)^{\frac{1}{2}}}\,$

SolutionIf $\displaystyle f=x^2y i-2xzj+2yzk\,$ , find i). $\displaystyle div f\,$ . ii). Evaluate $\displaystyle div [(x^2-y^2)i+2xyj+(y^2-2xy)k]\,$

Solution If $\displaystyle a_1i+a_2j+a_3k\,$ ,prove that $\displaystyle \nabla\cdot a=(\nabla a_1)\cdot i+(\nabla a_2)\cdot j+(\nabla a_3)\cdot k\,$

SolutionIf $\displaystyle a=(x+3y)i+(y-3z)j+(x-2z)k\,$ ,find $\displaystyle (a\cdot\nabla)a\,$

Solution Evaluate $\displaystyle \nabla\cdot(a\times r)r^n\,$ where a is a constant vector.

Solution Find $\displaystyle \nabla\times f \,$ or curl F, where

i). $\displaystyle F=x^2yi-2xzj+2yzk\,$ ii). $\displaystyle F=(x^2-y^2)i+2xyj+(y^2-2xy)k\,$

Solution Prove that $\displaystyle curl curl F=0\,$ where $\displaystyle F=zi+xj+yk\,$

Solution If $\displaystyle V=e^{xyz}(i+j+k)\,$ ,find $\displaystyle curl V\,$

Solution If $\displaystyle r=xi+yj+zk\,$ ,prove that i).$\displaystyle div r=3\,$

ii). If $\displaystyle r=xi+yj+zk\,$ show that $\displaystyle curl r=0\,$

Solution IF $\displaystyle f=xy^2i+2x^2yzj-3yz^2k\,$ ,Find $\displaystyle \mathrm{div}f,\mathrm{curl}f\,$ .What are their values at$\displaystyle (1,-1,1)\,$

Solution Find the $\displaystyle \mathrm{curl}\,$ of the vector $\displaystyle V=(x^2+yz)i+(y^2+zx)j+(z^2+xy)k\,$ at the point$\displaystyle (1,2,3)\,$

Solution If $\displaystyle f=(x+y+1)i+j+(-x-y)k\,$ ,prove that $\displaystyle f\cdot\mathrm{curl}f=0\,$

Solution a). Prove that vector $\displaystyle f=(x+3y)i+(y-3z)j+(x-2z)k\,$ is solenoidal.

b). Determine the constant 'a' so that the vector $\displaystyle f=(x+3y)i+(y-2z)j+(x+az)k\,$ is solenoidal.

Solution a). Show that the vector $\displaystyle f=(\sin y+z)i+(x\cos y-z)j+(x-y)k\,$ is irrational. b). Determine the constants 'a','b','c' so that the vector $\displaystyle f=(x+2y+az)i+(bx-3y-z)j+(4x+cy+2z)k\,$ is irrational.

Solution Prove that $\displaystyle \nabla\cdot(r^3 r)=6r^3\,$

Solution Prove $\displaystyle \mathrm{div}[r\nabla r^{-3}]=3r^{-4}\,$ or $\displaystyle \nabla\cdot[r\nabla(\frac{1}{r^3})]=\frac{3}{r^4}\,$

Solution If a is a constant vector,prove that $\displaystyle \mathrm{curl}\frac{a\times r}{r^3}=-\frac{a}{r^3}+\frac{3r}{r^5}(a\cdot r)\,$

Solution Show that $\displaystyle \nabla^2(\frac{x}{r^3})=0\,$

Solution Show that $\displaystyle \mathrm{div}\mathrm{grad}(r^m)=m(m+1)r^{m-2}\,$

Solution Evaluate $\displaystyle \mathrm{curl}\mathrm{grad}(r^m)\,$ where $\displaystyle r=|r|=|xi+yj+zk|\,$

Solution If $\displaystyle u=x^2-y^2+4z\,$ ,Show that $\displaystyle \nabla^{2} u=0\,$

Solution Show that $\displaystyle u=ax^2+by^2+cz^2\,$ satisfies Laplace equation $\displaystyle \nabla^2 u=0\,$

Solution If f and g are two scalar functions,prove that $\displaystyle \mathrm{div}(f\nabla g)=f\nabla^2 g+\nabla f\times\nabla g\,$

Solution Show that $\displaystyle \nabla\cdot(\nabla\times r)=0\,$ if $\displaystyle \nabla\times V=0\,$

Solution Prove that $\displaystyle \nabla^2(\frac{1}{r})=0\,$ ,where $\displaystyle r^2=x^2+y^2+z^2\,$

Solution Evaluate $\displaystyle \nabla^2(\frac{x}{r^2})\,$

Solution Prove that $\displaystyle V\times\mathrm{curl}V=\frac{1}{2}\nabla V^2-(V\cdot\nabla)V\,$

SolutionIf $\displaystyle v=v_1i+v_2j+v_3k\,$ ,prove that $\displaystyle \nabla\times v=\nabla v_1\times i+\nabla v_2\times j+\nabla v_3\times k\,$

Solution If r(P) be the vector from the origin O to a point P in the xy-plane,then show that the plane scalar field $\displaystyle u(P)=\log r\,$ satisfies the equation $\displaystyle \nabla^2 u=0\,$

Solution Prove that $\displaystyle \mathrm{div}(A\times r)=r\cdot\mathrm{curl}A\,$

Solution Prove that $\displaystyle \nabla\times(F\times r)=2F-(\nabla\cdot F)r+(r\cdot\nabla)F\,$

Solution If $\displaystyle u=e^{2x}+x^2z\,$ and $\displaystyle v=2z^2y-xy^2\,$ ,find $\displaystyle \mathrm{grad}(uv)\,$ at the point (1,0,2).

Solution Prove that $\displaystyle \mathrm{curl}[r\times(a\times r)]=3r\times a\,$ ,where a is a constant vector.

Solution Find the unit normal to the surface $\displaystyle z=x^2+y^2\,$ at the point (-1,-2,5).

Solution Find the directional derivative of $\displaystyle \phi=x^2yz+2xz^2\,$ at (1,-2,-1) in the direction of $\displaystyle 2i-j-2k\,$

Solution Calculate the maximum rate of change and the corresponding direction for the function $\displaystyle \phi=x^2y^3z^4\,$ at the point $\displaystyle 2i+3j-k\,$

Solution Find the equation of the tangent plane and normal to the surface $\displaystyle xyz=4\,$ at the point (1,2,2).

Solution Find the equation of the tangent line and normal plane to the curve of intersection of $\displaystyle x^2+y^2+z^2=1,x+y+z=1\,$ at (1,0,0).

Solution Find the angle between the curves $\displaystyle x^2+y^2+z^2=9,z=x^2+y^2-3\,$ at the point (2,-1,2).

Solution Find the constants a and b so that surfaces $\displaystyle ax^2-byz=(a+2)x\,$ will be orthogonal to the surface $\displaystyle 4x^2y+z^3=4\,$ at the point (1,-1,2).

### Line, Surface & Volume Integrals

Solution If $\displaystyle F=3xyi-y^2j\,$ ,Evaluate $\displaystyle \int_{C} F\cdot dr\,\,$ ,where C is the curve $\displaystyle y=2x^2\,$ in the xy-plane from (0,0)to (1,2).

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=(x^2+y^2)i-2xyj\,$ ,C is the rectangle in xy-plane bounded by $\displaystyle y=0,x=a,y=b,x=0\,$

Solution If $\displaystyle F=(2x+y)i+(3y-x)j\,$ ,evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where C is the curve in the xy-plane consisting of the straight line from $\displaystyle (0,0)\,$ to $\displaystyle (2,0)\,$ and then to $\displaystyle (3,2)\,$

Solution If $\displaystyle F=(3x^2+6y)i-14yzj+20xz^2k\,$ ,evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where C is the straight line joining $\displaystyle (0,0,0),(1,1,1)\,$

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=\cos y i-x\sin y j\,$ and C is the curve $\displaystyle y=\sqrt{1-x^2}\,$ in the xy-plane from $\displaystyle (1,0)\,$ to $\displaystyle (0,1)\,$

Solution Evaluate $\displaystyle \int F\cdot\,dr\,$ along the curve $\displaystyle x^2+y^2=1,z=1\,$ in the positive direction from (0,1,1) to (1,0,1),where $\displaystyle F=(2x+yz)i+xzj+(xy+2z)k\,$

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=x^2y^2i+yj\,$ and the curve c is $\displaystyle y^2=4x\,$ in the xy-plane from (0,0) to (4,4) where $\displaystyle r=xi+yj\,$

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=xyi+(x^2+y^2)j\,$ and C in the arc of the curve $\displaystyle y=x^2-4\,$ from (2,0) to (4,12).

Solution If $\displaystyle F=(2x^2+y^2)i+(3y-4x)j\,$ evaluate $\displaystyle \int F\cdot\,dr\,$ around the triangle ABC whose vertices are $\displaystyle A(0,0),B(2,0),C(2,1)\,$

Solution If $\displaystyle F=(2y+3)i+xzj+(yz-x)k\,$ .evaluate $\displaystyle \int_{C}F\cdot\,dr\,$ where C is the path consisting of the straight lines from (0,0,0) to (0,0,1) then to (0,1,1) and then to (2,1,1).

Solution If $\displaystyle A=(2y+3)i+xzj+(yz-x)k\,$ ,evaluate $\displaystyle \int_{C} A\cdot\,dr\,$ along the curve C.$\displaystyle x=2t^2,y=t,z=t^3\,$ from t=0 to t=1.

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=zi+xj+yk\,$ and C is the arc of the curve $\displaystyle r=\cos t i+\sin t j+tk\,$ from $\displaystyle t=0\,$ to $\displaystyle t=\pi\,$

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=yzi+zxj+xyk\,$ and C is the portion of the curve $\displaystyle r=a\cos t i+b\sin t j+ctk\,$ from $\displaystyle t=0\,$ to $\displaystyle t=\frac{\pi}{2}\,$

Solution Evaluate $\displaystyle \int_{C} F\cdot\,dr\,$ where $\displaystyle F=xyi+yzj+zxk\,$ and C is the arc of the curve $\displaystyle r=a\cos\theta i+a\sin\theta j+a\theta k\,$ from $\displaystyle \theta=0\,$ to $\displaystyle \theta=\frac{\pi}{2}\,$

Solution If $\displaystyle F=xyi-zj+x^2k\,$ ,evaluate $\displaystyle \int_{C} F\times\,dr\,$ where C is the curve $\displaystyle x=t^2,y=2t,z=t^3\,$ from t=0 to 1.

Solution Find the total work done in moving a particle in a force field given by $\displaystyle F=3xyi-5zj+10xk\,$ along the curve $\displaystyle x=t^2+1,y=2t^2,z=t^3\,$ from t=1 to t=2.

Solution Find the work done when a force $\displaystyle F=(x^2-y^2+x)i-(2xy+y)j\,$ moves a particle in xy-plane from (0,0) to (1,1) along the curve $\displaystyle y^2=x\,$

Solution Find the work done in moving a particle in a force field $\displaystyle F=3x^2i+(2xz-y)j+zk\,$ along the line joining (0,0,0) to (2,1,3).

Solution Find the work done in moving a particle once round a circle C in the xy-plane,if the circle has centre at the origin and radius 3 and when the force field is given by $\displaystyle F=(2x-y+z)i+(x+y-z^2)j+(3x-2y+4z)k\,$

Solution Find the circulation of F round the curve C where $\displaystyle F=yi+zj+zxk\,$ and C is the circle $\displaystyle x^2+y^2=1,z=0\,$

Solution Find the circulation of F round the curve C where $\displaystyle F=e^x\sin y i+e^x\cos y j\,$ and C is the rectangle whose vertices are $\displaystyle (0,0),(1,0),(1,\frac{\pi}{2}),(0,\frac{\pi}{2})\,$

Solution Evaluate $\displaystyle \iint_{S} (y^2z^2i+z^2x^2j+x^2y^2k)\cdot\,ds\,$ where S is the part of the sphere $\displaystyle x^2+y^2+z^2=1\,$ above the xy-plane.

Solution If $\displaystyle F=yi+(x-2xz)j-xyk\,$ ,evaluate $\displaystyle \iint_{S}(\nabla\times F)\cdot n\,dS\,$ where S is the surface of the sphere $\displaystyle x^2+y^2+z^2=a^2\,$ above the xy-plane.

Solution Evaluate $\displaystyle \iint_{S}(\mathrm{curl}F\cdot n\,dS\,$ where $\displaystyle F=yi+zj+xk\,$ and surface S in the part of the sphere $\displaystyle x^2+y^2+z^2=1\,$ above the xy-plane.

Solution Evaluate $\displaystyle \iint_S (y^2zi+z^2xj+x^2yk)\cdot dS\,$ where S is the surface of the sphere $\displaystyle x^2+y^2+z^2=a^2\,$ lying in the positive octant.

Solution Evaluate $\displaystyle \iint_{S}F\cdot n\,dS\,$ over the surface of the cylinder $\displaystyle x^2+y^2=9\,$ included in the first octant between z=0 and z=4 where $\displaystyle F=zi+xj-yzk\,$

Solution Evaluate $\displaystyle \iint_{S}F\cdot n\,dS\,$ where $\displaystyle F=2yxi-yzj+x^2k\,$ over the surface of the cube bounded by the coordinate planes and planes $\displaystyle x=a,y=a,z=a\,$

Solution Evaluate $\displaystyle \iint_{S}F\cdot n\,dS\,$ where $\displaystyle F=(x-z)i+(x^3+yz)j-3xy^2k\,$ and S in the surface of the cone $\displaystyle z=2-\sqrt{(x^2+y^2)}\,$ above the xy-plane.

Solution Evaluate $\displaystyle \iiint_V F\,dv\,$ where $\displaystyle F=xi+yj+zk\,$ and V is the region bounded by the surfaces $\displaystyle x=0,x=2,y=0,y=6,z=4,z=x^2\,$

Solution Evaluate $\displaystyle \iiint_V \phi\,dv\,$ where $\displaystyle \phi=45x^2y\,$ and V is the closed region bounded by the planes $\displaystyle 4x+2y+z=8,x=0,y=o,z=0\,$

Solution If $\displaystyle F=2xzi-xj+y^2k\,$ evaluate $\displaystyle \iiint_V F\,dV\,$ where V is the region bounded by the planes $\displaystyle x=y=z=0,x=y=z=1\,$

Solution Let r denote the position vector any point (x,y,z) measured from an origin O and let $\displaystyle r=|r|\,$ .Evaluate $\displaystyle \iint_S \frac{r}{|r|^3}\cdot dS\,$ where S denotes the sphere of radius a with center at the origin.

Solution Evaluate $\displaystyle \iint_S F\cdot n dS\,$ ,where $\displaystyle F=yi+2xj-zk\,$ and S in the surface of the plane 2x+y=6 in the first octant cut off by the plane z=4.

Solution If $\displaystyle F=(2x^2-3z)i-2xyj-4xk\,$ ,then evaluate i). $\displaystyle \iiint_V \nabla\times F dV \,$ ii). $\displaystyle \iiint_V F dV\,$ , where V is the region bounded by x=0,y=0,z=0 and $\displaystyle 2x+2y+z=4\,$

Solution Evaluate $\displaystyle \iiint_V (2x+y)dV\,$ where V is closed region bounded by the cylinder $\displaystyle z=4-x^2\,$ and the planes x=0,y=0,y=2 and z=0.

Solution Find the volume of the region common to the intersecting cylinders $\displaystyle x^2+y^2=a^2\,$ and $\displaystyle x^2+z^2=a^2\,$

### Green, Stokes & Gauss Divergence Theorems

• Green's Theorem in the plane - Relation between plane and line integrals

If R is a closed region in the xy-plane bounded by a simple closed curve C and if $\displaystyle \phi(x,y)\,$ and $\displaystyle \psi(x,y)\,$ are continuous functions having continuous partial derivatives in R,then $\displaystyle \oint (\psi dx+\phi dy)=\iint_R \left [ \frac{\partial\phi}{\partial x}-\frac{\partial\psi}{\partial y} \right ] dx dy\,$ where C is traversed in the positive (anti-clockwise) direction.

• Stokes Theorem - Relation between surface and line integrals

If F is any continuously differentiable vector point function and S is a surface bounded by a curve C,then $\displaystyle \oint F\cdot dr=\iint_S \mathrm{curl}F\cdot n dS\,$ where the unit normal n at any point of S is drawn in the direction in which a right-handed screw would move when rotated in the sense of description of C.

Solution If $\displaystyle F=(x^2-y^2)i+2xyj\,$ and $\displaystyle r=xi+yj\,$ ,find the value of $\displaystyle \int F\cdot dr\,$ around the rectangular boundary x=0,x=a,y=0,y=b.

SolutionEvaluate by Green's theorm in plane$\displaystyle \int_C(e^{-x}\sin y dx+e^{-x}\cos y dy)\,$ where C is the rectangle with vertices (0,0),$\displaystyle (\pi,0),(\pi,\frac{\pi}{2}),(0,\frac{\pi}{2})\,$ .

Solution Verify Green's theorm in plane for $\displaystyle \int_C[(x^2-xy^3)dx+(y^2-2xy)dy]\,$ where C is the square with vertices (0,0),(2,0),(2,2),(0,2).

Solution Verify Green's theorm in plane for $\displaystyle \oint_C[(3x^2-8y^2)dx+(4y-6xy)dy]\,$ where C is the boundary of the region defined by x=0,y=0,$\displaystyle x+y=1\,$

Solution Apply Green's theorm in the plane to evaluate $\displaystyle \int_C[(2x^2-y^2)dx+(x^2+y^2)dy]\,$ where C is the boundary of the curve enclosed by the x-axis and the semi-circle$\displaystyle y=(1-x^2)^{\frac{1}{2}}\,$

Solution Show that the area bounded by a simple closed curve C is given by $\displaystyle \frac{1}{2}\int_C(xdy-ydx)\,$ . Hence deduce that the area of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$

Solution Verify Green's theorm in a plane $\displaystyle \oint_C [(x^2-2xy)dx+(x^2y+3)dy]\,$ where C is the boundary of the region defined by $\displaystyle y^2=8x,x=2\,$

Solution Evaluate by Green's theorm $\displaystyle \oint_C[(\cos x\sin y-xy)dx+\sin x\cos y dy]\,$ where C is the circle $\displaystyle x^2+y^2=1\,$

Solution Evaluate by Green's theorm in the plane $\displaystyle \oint_C [(x^2-\cos h y)dx+(y+\sin x)dy]\,$ where C is the rectangle with vertices (0,0),($\displaystyle \pi\,$ ,0),($\displaystyle \pi\,$ ,1),(0,1).

Solution Evaluate $\displaystyle \oint_C [(y-\sin x)dx+\cos x dy]\,$ where C is the triangle whose vertices are (0,0),($\displaystyle \frac{\pi}{2}\,$ ,0),($\displaystyle \frac{\pi}{2}\,$ , by using Green's theorm in plane.

Solution Verify Green's theorm in the plane for $\displaystyle \oint_C[(xy+y^2)dx+x^2dy]\,$ where C is the closed curve of the region bounded by $\displaystyle y=x,y=x^2\,$

Solution Verify Green's theorm in plane for $\displaystyle \oint_C[(3x^2-8y^2)dx+(4y-6xy)dy]\,$ where C is the region bounded by the parabolas $\displaystyle y^2=x,y=x^2\,$

Solution Verify Stokes' theorm for $\displaystyle F=(x^2+y^2)i-2xyj\,$ taken round the rectangle bounded by $\displaystyle x=\pm a,y=0,y=b\,$

Solution Evaluate $\displaystyle \oint_C F\cdot dr\,$ by Stokes' theorm where $\displaystyle F=y^2 i+x^2 j-(x+z)k\,$ and C is the boundary of the triangle with vertices at (0,0,0),(1,0,0),(1,1,0).

Solution If $\displaystyle F=(2x^2+y^2)i+(3y-4x)j\,$ evaluate $\displaystyle \oint_C F\cdot dr\,$ where C is the boundary of the triangle with vertices (0,0),(2,0),(2,1)

Solution Verify Stokes'theorm for the function $\displaystyle F=x^2i+xyj\,$ integrated along the rectangle in the plane z=0,where sides are along the lines x=0,y=0,x=a and y=b.

Solution Evaluate by Stokes'theorm $\displaystyle \oint_C(e^x dx+2ydy-dz)\,$ where C is the curve $\displaystyle x^2+y^2=4,z=2\,$

Solution Evaluate by Stokes'theorm $\displaystyle \oint_C(\sin z dx-\cos x dy+\sin y dz)\,$ where C is the boundary of the rectangle $\displaystyle 0\le x \le\pi,0\le y \le 1,z=3\,$

Solution By converting into line integral,evaluate $\displaystyle \iint_S(\nabla\times A)\cdot n dS\,$ , where $\displaystyle A=(x-z)i+(x^3+yz)j-3xy^2k\,$ and S is the surface of the cone $\displaystyle z=2-\sqrt{(x^2+y^2)}\,$ above the xy-plane.

Solution By converting into a line integral evaluate $\displaystyle \iint_S(\nabla\times F)\cdot n dS\,$ where $\displaystyle F=(x^2+y-4)i+3xyj+(2xy+z^2)k\,$ and S is the surface of the paraboloid $\displaystyle z=4-(x^2+y^2)\,$ above the xy-plane.

Solution Evaluate $\displaystyle \iint_S(\nabla\times F)\cdot n dS\,$ where $\displaystyle F=(y-z+2)i+(yz+4)j-xzk\,$ and S is the surface of the cube $\displaystyle x=y=z=0,x=y=z=2\,$ above the xy-plane.

Solution Verify Stokes'theorm for $\displaystyle F=yi+zj+xk\,$ where S is the upper half surface of the sphere $\displaystyle x^2+y^2+z^2=1\,$ and C is its boundary.

Solution Verify Stokes'theorm for the vector $\displaystyle F=3yi-xzj+yz^2k\,$ where S is the surface of the paraboloid $\displaystyle 2z=x^2+y^2\,$ bounded by z=2 and C is its boundary.

Solution Verify Stokes'theorm for the function $\displaystyle F=zi+xj+yk\,$ where C is the unit circle in xy-plane bounding the hemisphere $\displaystyle z=\sqrt{(1-x^2-y^2)}\,$

Solution Apply Stokes'theorm to prove that $\displaystyle \int_C(ydx+zdy+xdz)=-2\sqrt{2}\pi a^2\,$ ,where C is the curve given by $\displaystyle x^2+y^2+z^2-2ax-2ay=0,x+y=2a\,$ and begins at the point (2a,0,0) and goes at first below the xy-plane.

Solution By Stokes' theorem,prove that $\displaystyle \mathrm{curl}\mathrm{grad}\phi=0\,$

Solution Evaluate $\displaystyle \iint_S F\cdot n dS\,$ where $\displaystyle F=axi+byj+czk\,$ and S is the surface of the sphere $\displaystyle x^2+y^2+z^2=1\,$

Solution Use divergence theorm to find $\displaystyle \iint_S F\cdot n dS\,$ for the vector $\displaystyle F=xi-yj+2zk\,$ over the sphere $\displaystyle x^2+y^2+(z-1)^2=1\,$

Solution If $\displaystyle F=ax i+by j+cz k\,$ where a,b,c are constants,show that $\displaystyle \iint_S(n\cdot F)dS=\frac{4}{3}\pi (a+b+c)\,$ ,S being the surface of the sphere $\displaystyle (x-1)^2+(y-2)^2+(z-3)^2=1\,$

Solution Find $\displaystyle \iint_S A\cdot n dS\,$ ,where $\displaystyle A=(2x+3z)i-(xz+y)j+(y^2+2z)k\,$ and S is the surface of the sphere having center at (3,-1,2) and radius 3 units.

Solution By using the Gauss Divergence theorm,evaluate $\displaystyle \iint_S(xdy dz+ydz dx+zdx dy)\,$ ,where S is the surface of the sphere $\displaystyle x^2+y^2+z^2=4\,$

Solution Apply divergence theorm to evaluate $\displaystyle \iint_S[(x+z)dy dz+(y+z)dz dx+(x+y)dx dy]\,$ where S is the surface of the sphere $\displaystyle x^2+y^2+z^2=4\,$

Solution If $\displaystyle F=4xzi-y^2 j+yzk\,$ ,then evaluate $\displaystyle \iint_s F\cdot n dS\,$ where S is the surface of the cube enclosed by x=0,x=1,y=0,y=1,z=0 and z=1.

Solution Evaluate $\displaystyle \iint F\cdot n dS\,$ ,where $\displaystyle F=4xyi+yzj-xzk\,$ and S is the surface of the cube bounded by the planes x=0,x=2,y=0,y=2,z=0,z=2.

Solution Apply Gauss'theorm to evaluate $\displaystyle \iint_S[(x^3-yz)dz dx-2x^2 y dz dx+zdx dy]\,$ over the surface S of a cube bounded by the coordinate planes and the planes x=y=z=a.

Solution Apply Gauss'theorm to show that $\displaystyle \iint_S(x^3-yz)i-2x^2yj+2k]\cdot n dS=\frac{a^5}{3}\,$ ,where S denotes the surface of the cube bounded by the planes $\displaystyle x=0,x=a,y=0,y=a,z=0,z=a\,$

Solution Evaluate $\displaystyle \iint_s[x^2 dy dz+y^2 dz dx+2z(xy-x-y)dx dy]\,$ where S is the surface of the cube $\displaystyle 0\le x\le 1,0\le y\le 1,0\le z\le 1\,$

Solution Find the value of $\displaystyle \iint_S(F\times\nabla\phi)\cdot n dS\,$ where $\displaystyle F=x^2 i+y^2 j+z^2 k,\phi=xy+yz+zx,S:x=\pm 1,y=\pm 1,z=\pm 1\,$

Solution Evaluate $\displaystyle \iint_S F\cdot n dS\,$ ,where $\displaystyle F=xi-yj+(z^2-1)k\,$ and S is the closed surface bounded by the planes $\displaystyle z=0,z=1\,$ and the cylinder $\displaystyle x^2+y^2=4\,$ by the application of Gauss'theorm.

Solution Use Gauss' theorem to evaluate the integral $\displaystyle \iint_S F\cdot n dS\,$ of the vector field $\displaystyle F=xy^2 i+y^3 j+y^2 zk\,$ through the closed surface formed by the cylinder $\displaystyle x^2+y^2=9\,$ and the plane $\displaystyle z=0,z=2\,$

Solution Use Gauss divergence theorem to find $\displaystyle \iint_S F\cdot n dS\,$ where $\displaystyle F=2x^2 y i-y^2 j+4xz^2 k\,$ and S is the closed surface in the first octant bounded by $\displaystyle y^2+z^2=9,x=2\,$

Solution Evaluate $\displaystyle \iint_S(zx^2 dx dy+x^3 dy dz+yx^2 dz dx)\,$ where S is the closed surface consisting of the cylinder $\displaystyle x^2+y^2=4\,$ and the circular discs $\displaystyle z=0,z=3\,$

Solution If $\displaystyle F=\nabla\phi,\nabla^2\phi=-4\pi \rho\,$ show that $\displaystyle \iint_S F\cdot n dS=-4\pi\iiint_V\rho dV\,$

SolutionIf $\displaystyle \phi\,$ is harmonic in V,then $\displaystyle \iint_S \frac{\partial\phi}{\partial n} dS=\iiint_V\nabla^2\phi dV\,$ where S is the surface enclosing V.

Solution Prove that i). $\displaystyle \iiint_V\nabla\phi\cdot A dV=\iint_S\phi\cdot n dS-\iiint_V\phi\nabla\cdot A dV\,$ ii).Prove that $\displaystyle \iiint_V F\cdot\mathrm{curl}G dV=\iint_S G\times F\cdot dS+\iiint_V G\cdot\mathrm{curl}F dV\,$

Solution Show that for any closed surface S, i). $\displaystyle \iint_S n dS=0\,$ ii).$\displaystyle \iint_S r\times n dS=0\,$ iii).$\displaystyle \iint_S(\nabla\phi)\times n dS=0\,$

Solution If V is the volume of a region T bounded by a surface S,then prove that $\displaystyle V=\iint_S x dy dz=\iint_S y dz dx=\iint_S z dx dy\,$

Solution Evaluate $\displaystyle \iint_S(\nabla\times F)\cdot n dS\,$ ,where $\displaystyle F=(x-z)i+(x^3+yz)j-3xy^2k\,$ and S is the surface of the cone $\displaystyle z=2-\sqrt{(x^2+y^2)}\,$ above the xy plane.

Solution Evaluate $\displaystyle \iint_S(x^3 dy dz+y^3 dz dx+z^3 dx dy)\,$ by converting the surface integral into a volume integral.Here,S is the surface of the sphere $\displaystyle x^2+y^2+z^2=1\,$

Solution Evaluate with the help of divergence theorm the integral $\displaystyle \iint_S[xz^2 dy dz+(x^2y-z^3)dz dx+(2xy+y^2z)dx dy]\,$ , where S is the entire surface of the hemispherical region bounded by $\displaystyle z=\sqrt{(a^2-x^2-y^2)},z=0\,$

Solution Evaluate $\displaystyle \iint_S(ax^2+by^2+cz^2)dS\,$ over the sphere $\displaystyle x^2+y^2+z^2=1\,$ using the divergence theorm.

Solution Compute $\displaystyle \iint_S(a^2x^2+b^2y^2+c^2z^2)^{\frac{1}{2}}dS\,$ over the ellipsoid $\displaystyle ax^2+by^2+cz^2=1\,$

Solution Compute $\displaystyle \iint_S(a^2x^2+b^2y^2+c^2z^2)^{\frac{-1}{2}}dS\,$ over the ellipsoid $\displaystyle ax^2+by^2+cz^2=1\,$

Solution Evaluate$\displaystyle \iint_S F\cdot n dS\,$ over the entire surface of the region above the xy plane bounded by the cone $\displaystyle z^2=x^2+y^2\,$ and the plane z=4,if $\displaystyle F=xi+yj+z^2k\,$

Solution By using the Gauss divergence theorm evaluate $\displaystyle \iint_S(xi+yj+z^2k)\cdot n dS\,$ ,where S is the closed surface bounded by the cone $\displaystyle x^2+y^2=z^2\,$ and the plane z=1.

Solution Evaluate $\displaystyle \iint_S(\nabla\times F)\cdot n dS\,$ where $\displaystyle A=[xye^z+\log(z+1)-\sin x]k\,$ and S is the surface of the sphere $\displaystyle x^2+y^2+z^2=a^2\,$ above the xy plane.

Solution Evaluate $\displaystyle \iint_S xyz dS\,$ where S is the surface of the sphere $\displaystyle x^2+y^2+z^2=a^2\,$

Solution By transforming to a triple integral,evaluate $\displaystyle \iint_S(x^3 dy dz+x^2y dz dx+x^2z dx dy)\,$ where S is the closed surface bounded by the planes $\displaystyle z=0,z=b\,$ and the cylinder $\displaystyle x^2+y^2=a^2\,$

Solution Evaluate $\displaystyle \iint_S \frac{1}{p}dS\,$ where S is the surface of the ellipsoid $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\,$ and p is the perpendicular drawn from the origin to the tangent plane at $\displaystyle (x,y,z)\,$

Solution Show that $\displaystyle \iint_S(x^2i+y^2j+z^2k)\cdot n dS\,$ vanishes where S denotes the surface of the ellipsoid $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\,$

Solution Verify the divergence theorm theorm for $\displaystyle F=4xzi-y^2j+yzk\,$ taken over the cube bounded by $\displaystyle x=0,x=1,y=0,y=1,z=0,z=1\,$

Solution Evaluate $\displaystyle \iint_S(x^3 dy dz+y^3 dz dx)\,$ where S is the surface of hte sphere $\displaystyle x^2+y^2+z^2=a^2\,$

Solution Show that the vector field $\displaystyle F=(2xy^2+yz)i+(2x^2y+xz+2yz^2)j+(2y^2z+xy)k\,$ is conservative.

Solution Show that the vector field defined by $\displaystyle F=(2xy-z^3)i+(x^2+z)j+(y-3xz^2)k\,$ is conservative and find the scalar potential of F.

Solution Show that the vector field F given by $\displaystyle F=(y+\sin z)i+xj+x\cos z k\,$ is conservative,find its scalar potential.

Solution Show that $\displaystyle F=xi+yj+zk\,$ is conservative and find $\displaystyle \phi\,$ such that $\displaystyle F=\nabla\phi\,$

Solution Prove that $\displaystyle F=|r|^2 r\,$ is conservative and find its scalar potential.

Solution Show that $\displaystyle (y^2z^3\cos x-4x^3z)dx+2z^3 y\sin x dy+(3y^2z^2\sin x-x^4)dz\,$ is an exact differential of some function $\displaystyle \phi\,$ and find this function.

Solution Show that $\displaystyle (2x\cos y+z\sin y)dx+(xz\cos y-x^2\sin y)dy+x\sin y dz=0\,$ is an exact differential and hence solve it.

Solution Evaluate $\displaystyle \int_C[2xyz^2 dx+(x^2z^2+z\cos yz)dy+(2x^2yz+y\cos yz)dz\,$ where C is any path from $\displaystyle (0,0,0)\,$ to $\displaystyle 1,\frac{\pi}{4},2)\,$

Solution If $\displaystyle F=\cos y i-x\sin y j\,$ , evaluate $\displaystyle \int_C F\cdot dr\,$ where C is the curve $\displaystyle y=\sqrt{(1-x^2)}\,$ in the xy plane from $\displaystyle (1,0)\,$ to $\displaystyle (0,1)\,$

Solution Evaluate $\displaystyle \int_C[yz dx+(xz+1)dy+xy dz]\,$ ,where C is any path from $\displaystyle (1,0,0)\,$ to $\displaystyle (2,1,4)\,$

Solution A vector field is given by $\displaystyle F=(x^2+xy^2)i+(y^2+x^2y)j\,$ Show that the field is irrational and obtain its scalar potential.

Solution Show that the vector field F given by $\displaystyle F=(x^2-yz)i+(y^2-zx)j+(z^2-xy)k\,$ is irrotational. Find a scalar $\displaystyle \phi\,$ such that $\displaystyle F=\nabla\phi\,$

Solution Show that the vector function $\displaystyle F=(\sin y+z\cos x)i+(x\cos y+\sin z)j+(y\cos z+\sin x)k\,$ is irrotational and find the scalar function $\displaystyle \phi\,$ such that $\displaystyle F=\nabla\phi\,$

## Multiple Integrals

solution $\displaystyle \int_0^2 \int_0^1 (2x+y)^8 dx dy\,$

solution Evaluate $\displaystyle \int_0^2\int_1^2(x^2+y^2)dx dy\,$

solution Evaluate $\displaystyle \int_0^1\int_1^2(x^2+y^2)dx dy\,$

solution $\displaystyle \int_0^3\int_1^2 xy(x+y)dx dy\,$

solution $\displaystyle \int_0^a\int_0^b(x^2+y^2)dx dy\,$

solution $\displaystyle \int_1^2\int_3^4 \frac{1}{(x+y)^2}dx dy\,$

solution $\displaystyle \int_1^4\int_{0}^{\sqrt{4-x}}xydx dy\,$

solution $\displaystyle \int_1^2\int_{x}^{x\sqrt{3}}xydx dy\,$

solution $\displaystyle \int_1^2\int_1^x xy^2dx dy\,$

solution $\displaystyle \int_{0}^{\frac{\pi}{4}}\int_{0}^{\frac{\pi}{2}}\sin(x+y)dx dy\,$

solution $\displaystyle \int_0^a\int_{0}^{\sqrt{a^2-x^2}}y^3dy dx\,$

solution $\displaystyle \int_0^1\int_{\sqrt{y}}^{2-y}x^2dx dy\,$

solution $\displaystyle \int_0^2\int_{x^2}^{2x}(2x+3y)dy dx\,$

solution $\displaystyle \int_0^a\int_{0}^{\sqrt{a^2-x^2}}xydx dy\,$

solution $\displaystyle \int_0^1\int_{0}^{1-x}(x^2+y^2)dy dx\,$

solution $\displaystyle \int_0^a\int_{\frac{x^2}{a}}^{2a-x}xydy dx\,$

solution $\displaystyle \int_0^1\int_{-\sqrt{y}}^{\sqrt{y}}dx dy+\int_1^9\int_{\frac{y-3}{2}}^{\sqrt{y}}dx dy\,$

solution $\displaystyle \int_{0}^{2a}\int_{\frac{y^2}{4a}}^{3a-y}(x^2+y^2)dx dy\,$

solution Evaluate $\displaystyle \iint_R xydx dy\,$ where R is the positive quadrant of the circle $\displaystyle x^2+y^2=a^2\,$

solution Evaluate the double integral $\displaystyle \iint xy(x+y)dx dy\,$ over the region bounded by the curves $\displaystyle y=x,y=x^2\,$

solution Evaluate $\displaystyle \int_{0}^{2a}\int_{0}^{\sqrt{2ax-x^2}}(x^2+y^2)dx dy\,$ by changing into polar coordinates.

solution Evaluate $\displaystyle \iint xy(x^2+y^2)^{\frac{n}{2}}dx dy\,$ over the positive quadrant of the circle $\displaystyle x^2+y^2=a^2\,$ supposing n+3>0.

solution Find $\displaystyle \iint_R xydx dy\,$ where R is the region bounded by $\displaystyle x=1,x=2,y=0,xy=1\,$

solution Evaluate $\displaystyle \int_{1}^{\log 8}\int_{0}^{\log y}e^{x+y}dx dy\,$

solution Evaluate $\displaystyle I=\iint_D(x^2+y^2)dx dy\,$ where D is bounded by $\displaystyle y=x\,$ and $\displaystyle y^2=4x\,$

solution $\displaystyle \iint_D(4xy-y^2)dx dy\,$ where D is the reactangle bounded by $\displaystyle x=1,x=2,y=0,y=3\,$

solution $\displaystyle \iint_D(x^2+y^2)dx dy\,$ where D is the region bounded by $\displaystyle y=x,y=2x,x=1\,$ in the first quadrant.

solution $\displaystyle \iint_D(1+x+y)dx dy\,$ where D is the region bounded by the lines $\displaystyle y=-x,x=\sqrt{y},y=2,y=0\,$

solution $\displaystyle \iint_D xy dx dy\,$ where D is the domain bounded by the parabola $\displaystyle x^2=4ay\,$ ,the ordinates $\displaystyle x=a\,$ and x-axis.

solution Evaluate $\displaystyle \iint (x-y)dx dy\,$ over the region between the line $\displaystyle x=y\,$ and the parabola $\displaystyle y=x^2\,$

solution Find the area bounded by the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$

solution $\displaystyle I_D=\iint_D x^3y dx dy\,$ where D is the region enclosed by the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ in the first quadrant.

solution Find by double integration,the area which lies inside the cardoid $\displaystyle r=a(1+\cos\theta)\,$ and outside the circle r=a.

solution Find the area in the XY-plane bounded by the lemniscate $\displaystyle r^2=a^2\cos 2\theta\,$

solution Find the area bounded by the curves $\displaystyle y^2=x^3\,$ and $\displaystyle x^2=y^3\,$

solution Find the area of the domains $\displaystyle 3x=4-y^2,x=y^2\,$ in the XY-plane.

solution Find the area of the region bounded by the parabola $\displaystyle y^2=4ax\,$ and the straight line $\displaystyle x+y=3a\,$ in the XY-plane.

solution Find the area common to the parabolas $\displaystyle y^2=4a(x+a),y^2=4b(b-x)\,$

solution Find the area of the domains $\displaystyle x=y-y^2,x+y=0\,$ in the XY-plane.

solution Find the area of the domains $\displaystyle 3y^2=25x,5x^2=9y\,$ in the XY-plane.

solution Find the mass,coordinates of the centre of gravity and moments of inertia relative to x-axis,y-axis and origin of a reactangle $\displaystyle 0\le x\le 4,0\le y\le 2\,$ having mass density xy.

solution Find the volume of tetrahedron in space cut from the first octant by the plane $\displaystyle 6x+3y+2z=6\,$

solution Calculate the volume of a solid whose base is in a xy-plane and is bounded by the parabola $\displaystyle y=4-x^2\,$ and the straight line $\displaystyle y=3x\,$ ,while the top of the solid is in the plane $\displaystyle z=x+4\,$ .

solution Find the moment of inertia of the area bounded by the circle $\displaystyle x^2+y^2=a^2\,$ in the first quadrant,assume the surface density of 1.

solution A plane lamina of non uniform density is in the form of a quadrant of the ellipse $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\,$ .If the density at any point (x,y)be kxy,where K is a constant,find the coordinates of the centroid of the lamina.

solution Determine the volume of the space below the paraboloid $\displaystyle x^2+y^2+z-4=0\,$ and above the square in the xy-plane with vertices at $\displaystyle (0,0),(0,1),(1,0),(1,1)\,$ .

solution Find the volume of the solid under the surface $\displaystyle az=x^2+y^2\,$ and whose base R is the circle $\displaystyle x^2+y^2=a^2\,$ .

solution Find the volume enclosed by the coordinate planes and that portion of the plane $\displaystyle x+y+z=1\,$ which lies in the first quadrant.

solution A circular hole of radius b is made centrally through a sphere of radius a.Find the volume of the remaining portion of the sphere.

solution Find the volume of the region bounded by the paraboloids $\displaystyle z=x^2+y^2\,$ and $\displaystyle z=\frac{6-\frac{x^2+y^2}{2}}\,$ .

solution Find the volume of the region bounded by the paraboloid $\displaystyle z=x^2+y^2\,$ and the plane z=4.

solution Find the volume of the solid enclosed by the ellipsoid $\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\,$

solutionFind the volume of the region in space bounded by the surface $\displaystyle z=1-(x^2+y^2)\,$ on the sides by the planes $\displaystyle x=0,y=0,x+y=1\,$ and below by the plane z=0.

solution Evaluate $\displaystyle \int_0^1\int_0^x\int_{0}^{x+y}(x+y+z)dz dy dx\,$

solution Find the volume bounded by the ellipsoidic paraboloids $\displaystyle z=x^2+9y^2\,$ and $\displaystyle z=18-x^2-9y^2\,$

solution Find the total mass of the region in the cube $\displaystyle 0\le x\le 1,0\le y\le 1,0\le z\le 1\,$ with density at any point given by $\displaystyle xyz\,$

solution Find the mass,centroid of the tetrahedron bounded by the coordinate planes and the plane $\displaystyle \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\,$

solution Evaluate $\displaystyle \int_0^2\int_1^z\int_{0}^{yz} xyz dx dy dz\,$

solution If the radius of the base and altitude of a right circular cone are given by a and h respectively,express its volume as a tripple integral and evaluate it using cylindrical coordinates.

solution Evaluate $\displaystyle \int_0^a\int_0^x\int_{0}^{y+x} e^{x+y+z} dz dy dx\,$

solution Find the volume bounded by the sphere $\displaystyle x^2+y^2+z^2=a^2\,$
solution Evaluate $\displaystyle \iiint xyz dx dy dz\,$ over the positive octant of the sphere $\displaystyle x^2+y^2+z^2=a^2\,$

solution Assuming $\displaystyle \rho(x,y,z)=1\,$ ,find the centroid of the portion of the sphere $\displaystyle x^2+y^2+z^2=a^2\,$ in the first octant.

solution Find the mass and moment of inertia of a sphere of radius 'a' with respect to a diameter if the density is proportional to the distance from the center.

solution Evaluate $\displaystyle \iiint_E y^2 x^2 dV\,$ Where E is the region bounded by the paraboloid $\displaystyle x= 1 - y^2 - z^2 \,$ and the plane $\displaystyle \,x = 0$

solution $\displaystyle \int_{-2}^{2}\int_{0}^{\sqrt{4-y^2}}\int_{-\sqrt{4-x^2-y^2}}^{\sqrt{4-x^2-y^2}} y^2\sqrt{x^2+y^2+z^2} dzdxdy \,$ Evaluate using spherical coordinates

solution Evaluate $\displaystyle \iint_{R}(x+y)e^{x^2-y^2}dA \,$ Where $\displaystyle _{R}\,$ is the rectangle enclosed by the lines $\displaystyle x - y = 0 , x - y = 2 , x + y = 0 , x + y = 3 \,$