Double integrals over regions in the plane, iterated integrals, and changes of order of integration.
These practice problems are about integrating a function of two variables over a region in the plane. You will describe the region, write an iterated integral, change the order of integration, pass to polar coordinates or another change of variables, and use the integral for area, mass, and surface area. Step-by-step solutions show both the region and the iterated integral.
Iterated Integrals is the place to start: the region is given, and you integrate in a stated order. Changing the Order of Integration is the chapter you need when one order has no elementary antiderivative and the other does. Polar Coordinates and Change of Variables covers disks, sectors, and regions where a Jacobian simplifies the integrand. Applications turns the same integrals into area, mass, centroids, and surface area.
A definite integral in one variable should already be comfortable. If it is not, use Definite Integrals first, and Applications of Definite Integrals for area and volume with a single integral, which is the easier geometric set. The change-of-variables problems here are the hardest multivariable exercises on the site.
Problem 1
HardIterated Integrals
Starting from the definition, calculate the following integral:
$$ \iint_{\substack{0 \leqslant x \leqslant 1 \\ 0 \leqslant y \leqslant 1}} xy \, dx \, dy. $$
Starting from the definition, calculate the following integral:
$$ \iint_{\substack{a \leqslant x \leqslant b \\ c \leqslant y \leqslant d}} x^2 y^2 \, dx \, dy. $$
Starting from the definition, calculate the following integral:
$$ \iint_{\substack{a \leqslant x \leqslant b \\ c \leqslant y \leqslant d}} e^{x+y} \, dx \, dy. $$
Form the lower sum $\underline{S}$ and upper sum $\overline{S}$ for the function $z(x,y)=x^2+y^2$ on the region $1\leqslant x\leqslant 2,\ 1\leqslant y\leqslant 3$, dividing this region into rectangles by the lines
$$ x=1+\frac{i}{n}, \qquad y=1+\frac{2j}{n} \qquad (i,j=0,1,\dots,n). $$
Find the limit value of these sums as $n\to\infty$.
Verify the following relations:
$$ 1^{\circ}\quad 8(5-\sqrt2)\pi < \iint_D (x+y+100)\,dx\,dy < 8(5+\sqrt2)\pi $$
$$ 2^{\circ}\quad 36\pi < \iint_D (x^2+4y^2+9)\,dx\,dy < 76\pi $$
where the region $D$ is bounded by the circle $x^2+y^2=4$.
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is a triangle with vertices $O(0,0)$, $A(1,0)$, $B(1,1)$.
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is a parallelogram with vertices $A(1,2)$, $B(2,4)$, $C(2,7)$, $D(1,5)$.
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is a circle:
$1^{\circ}\ x^2+y^2\leqslant1$; $2^{\circ}\ x^2+y^2\leqslant x$.
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is an annulus: $4<x^2+y^2<9$.
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is bounded by the lines $y=x$ and $y=\sqrt{4x-x^2}$.
Write the limits of integration for the double integral $\iint_D z(x,y)\,dx\,dy$ for both possible orders of integration. The region $D$ is defined by the inequality $|x|+|y|<1$.
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is:
$1^{\circ}$ Circle $x^2+y^2\leqslant a^2$; $2^{\circ}$ Circular annulus $a^2<x^2+y^2<b^2$.
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is the circle $x^2+y^2<ax$ where $a>0$.
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is bounded by the curve $r=1-\varphi^2$ where $r>0$.
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is a rectangle with vertices $O(0,0)$, $A(1,0)$, $B(1,1)$, $C(0,1)$.
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is bounded by the lines $x=2$, $y=x$, and $y=x\sqrt3$.
In the double integral $\iint_D z(x,y)\,dx\,dy$, convert to polar coordinates $\varphi$ and $r$ and write the limits of integration for both possible orders. The region $D$ is bounded by the curve $(x^2+y^2)^2=a^2(x^2-y^2)$ where $x>0$.
Assuming that $\varphi$ and $r$ are polar coordinates, change the order of integration:
$$ \int_{-\pi/2}^{\pi/2} d\varphi \int_0^{\cos\varphi} z(\varphi,r)\,dr. $$
Assuming that $\varphi$ and $r$ are polar coordinates, change the order of integration:
$$ \int_0^{\pi/2} d\varphi \int_0^{a\sqrt{\sin2\varphi}} z(\varphi,r)\,dr \qquad (a>0). $$
Assuming that $\varphi$ and $r$ are polar coordinates, change the order of integration:
$$ \int_0^{a} d\varphi \int_0^{\varphi} z(\varphi,r)\,dr \qquad (0<a<\pi). $$
Calculate the following integral:
$$ \iint_D xy^2\,dx\,dy, $$
where the region of integration is bounded by the parabola $y^2=2x$ and the line $x=\frac12$.
Calculate the following integral:
$$ \iint_D \frac{x}{2}\,dx\,dy, $$
where the region $D$ is bounded by the lines $x=0$, $x=2+\sin y$, $y=0$, $y=2\pi$.
Calculate the following integral:
$$ \iint_D y^2\,dx\,dy, $$
where the region $D$ is bounded by the $x$-axis and the first arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$.
Find:
$$ \lim_{\alpha\to0^+} I(a), \qquad \text{where } I(a)=\iint_D (x+y)^{-a}\,dx\,dy, $$
and $D$ is the triangular region $x\geqslant0$, $y\geqslant0$, $\alpha<x+y<1$, with $\alpha\to0^+$.
Calculate the following integral:
$$ \iint_D \frac{dx\,dy}{\sqrt{x}+\sqrt{y}}, $$
where $D$ is the triangle $0\leqslant y\leqslant x\leqslant1$ (vertices $(0,0)$, $(1,0)$, $(1,1)$).
Calculate the following integral:
$$ \iint_D x^2y\sqrt{1-x^3-y^3}\,dx\,dy, $$
where $D$ is the region $x\geqslant0$, $y\geqslant0$, $x^3+y^3\leqslant1$.
By converting to polar coordinates, calculate the following integral:
$$ \iint_D xy\,dx\,dy, $$
where the region $D$ is bounded by the $x$-axis and the arcs of the circles $x^2+y^2=1$ and $x^2+y^2-2x=0$.
Calculate the following integral:
$$ \iint_D (x^2+y^2)\,dx\,dy, $$
where $D$ is the disk $x^2+y^2\leqslant2ay$ (radius $a$, centered at $(0,a)$, tangent to the $x$-axis at the origin).
By converting to polar coordinates, calculate the following integral over each of the two regions described below:
$$ \iint_D \sqrt{a^2-x^2-y^2}\,dx\,dy. $$
$1^{\circ}$ $D$ is bounded by the circle $x^2+y^2=a^2$ and the lines $y=x$ and $y=x\sqrt3$.
$2^{\circ}$ $D$ is the disk $x^2+y^2-ax\leqslant0$.
By converting to polar coordinates, calculate the following integral:
$$ \iint_D \frac{dx\,dy}{(x^2+y^2)\left(1+\sqrt[8]{x^2+y^2}\right)}, $$
where $D$ is the region $x^2-y^2<0$, $1<x^2+y^2<4$.
Calculate the following integral:
$$ \iint_D \sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}\,dx\,dy, $$
where $D$ is the region $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}\leqslant1$.
Calculate the following integral:
$$ \iint_D \sqrt{4-\left(\frac xa\right)^2-\left(\frac yb\right)^2}\,dx\,dy, $$
where $D$ is the first-quadrant region bounded by the ellipses $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ and $\dfrac{x^2}{(2a)^2}+\dfrac{y^2}{(2b)^2}=1$.
Calculate the following integral:
$$ \iint_D \sqrt{\sqrt{x}+\sqrt{y}}\,dx\,dy, $$
where $D$ is the region bounded by the coordinate axes and the curve $\sqrt{x}+\sqrt{y}=1$.
Calculate the following integral:
$$ \iint_D \arctan\frac{y}{x}\,dx\,dy, $$
where $D$ is the part of the annulus $1\leqslant x^2+y^2\leqslant9$ satisfying $y>\dfrac{x}{3}$ and $y>x\sqrt3$.
Calculate the following integral over each of the two regions described below:
$$ \iint_D |\cos(x+y)|\,dx\,dy. $$
$1^{\circ}$ $D$ is the square $0\leqslant x\leqslant\pi$, $0\leqslant y\leqslant\pi$.
$2^{\circ}$ $D$ is the triangle $0\leqslant y\leqslant x\leqslant\pi$.
Calculate the value of the integral:
$$ \iint_D (y-x)\,dx\,dy, $$
where the region $D$ is bounded by the lines
$$ y=x+1,\quad y=x-3,\quad y=-\frac13x+\frac79,\quad y=-\frac13x+5, $$
using the substitution $u=y-x$, $v=y+\frac13x$.
By introducing new variables $u$ and $v$ in place of $x$ and $y$, determine the limits of integration for:
$$ \int_a^b dx \int_{\alpha x}^{\beta x} z(x,y)\,dy \qquad (0<a<b;\ 0<\alpha<\beta), $$
using the substitution $u=x$, $v=\dfrac{y}{x}$.
By introducing new variables $u$ and $v$ in place of $x$ and $y$, determine the limits of integration for:
$$ \int_0^2 dx \int_{1-x}^{2-x} z(x,y)\,dy, $$
using the substitution $u=x+y$, $v=x-y$.
By introducing new variables $u$ and $v$ in place of $x$ and $y$, determine the limits of integration for:
$$ \iint_D z(x,y)\,dx\,dy, $$
where $D$ is bounded by $\sqrt x+\sqrt y=\sqrt a$, $x=0$, $y=0$ (with $a>0$), using the substitution $x=u\cos^4v$, $y=u\sin^4v$.
By introducing new variables $u$ and $v$ in place of $x$ and $y$, determine the limits of integration for:
$$ \iint_D z(x,y)\,dx\,dy, $$
where $D$ is the triangle bounded by $x=0$, $y=0$, $x+y=a$, using the substitution $x=\dfrac{u(a-v)}{a}$, $y=\dfrac{uv}{a}$.
How should the change of variables be performed so that the curvilinear quadrilateral bounded by the curves $xy=1$, $xy=2$, $x-y+1=0$, $x-y-1=0$ (where $x>0$, $y>0$) is mapped to a rectangle whose sides are parallel to the coordinate axes?
In the integral $\iint_D z(x,y)\,dx\,dy$, the region $D$ is the quarter disk $x^2+y^2\leqslant a^2$, $x>0$, $y>0$.
$1^{\circ}$ Map this region to a rectangle by a change of variables.
$2^{\circ}$ Map this region to an isosceles right triangle by a change of variables.
Calculate the following integral:
$$ \iint_D y^3\sqrt{1-x^2-y^4}\,dx\,dy, $$
where $D$ is the region $x^2+y^4\leqslant1$, $x\geqslant0$, $y\geqslant0$.
Let $z(x,y)$ be continuous in a neighborhood of the origin. Prove that:
$$ \lim_{\varrho\to0}\frac{1}{\pi\varrho^2}\iint_{x^2+y^2\leqslant\varrho^2} z(x,y)\,dx\,dy = z(0,0). $$
Let $\varphi$ be continuous and positive on $[-c,c]$. Prove that:
$$ \iint_{x^2+y^2<c^2} \frac{a\varphi(x)+b\varphi(y)}{\varphi(x)+\varphi(y)}\,dx\,dy = \frac{(a+b)c^2\pi}{2}. $$
Show that for the function $f(x,y)=\dfrac{x^2-y^2}{(x^2+y^2)^2}$ on the unbounded square $x\geqslant1,\,y\geqslant1$, the two iterated integrals:
$$ I_1=\int_1^\infty\left(\int_1^\infty f(x,y)\,dy\right)dx, \qquad I_2=\int_1^\infty\left(\int_1^\infty f(x,y)\,dx\right)dy $$
are not equal — and explain why this does not contradict Fubini's theorem.
Calculate the following improper integral, and determine for which $\alpha\in\mathbb R$ it converges:
$$ \iint_D \frac{dx\,dy}{(1-x^2-y^2)^{\alpha}}, $$
where $D$ is the region $x^2+y^2<1$.
Determine whether the following integral converges, and if so, evaluate it:
$$ \iint_D \ln\sqrt{x^2+y^2}\,dx\,dy, $$
where $D$ is the region $x^2+y^2<a^2$.
Determine whether the following integral converges:
$$ \iint_D \frac{\sin(x^2+y^2)}{(x^2+y^2)^{3/2}}\,dx\,dy, $$
where $D$ is the region $x^2+y^2<a^2$.
Can the number $m$ be chosen so that the improper integral:
$$ \iint \frac{dx\,dy}{\sqrt{(x^2+y^2)^m}}, $$
taken over the entire $xOy$ plane, converges?
Using generalized polar coordinates $x=ar\cos^\alpha\varphi$, $y=br\sin^\alpha\varphi$, find the area of the region bounded by:
$1^\circ$ $\left(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}\right)^2=\dfrac{x^2}{h^2}+\dfrac{y^2}{k^2}$;
$2^\circ$ $\dfrac{x^3}{a^3}+\dfrac{y^3}{b^3}=1$ ($x\geqslant0,\ y\geqslant0$).
Using generalized polar coordinates, find the area of the region (in the first quadrant) bounded by:
$$ \frac{x^3}{a^3}+\frac{y^3}{b^3} = \frac{x^2}{h^2}+\frac{y^2}{k^2}, \qquad x\geqslant0,\ y\geqslant0. $$
Using an appropriate change of variables, calculate the area bounded by the curve:
$$ \left(\frac xa+\frac yb\right)^2 = \frac xa-\frac yb, \qquad y>0. $$
Using an appropriate change of variables, calculate the area of the loop bounded by the curve:
$$ \left(\frac xa+\frac yb\right)^3 = \frac{xy}{c^2}. $$
Using an appropriate change of variables, calculate the area bounded by the curve:
$$ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2 = \frac{xy}{c^2}. $$
Using an appropriate change of variables, calculate the area bounded by the curves:
$$ \sqrt{\frac xa}+\sqrt{\frac yb}=1; \qquad \sqrt{\frac xa}+\sqrt{\frac yb}=2; \qquad \frac xa=\frac yb; \qquad 4\frac xa=\frac yb, $$
where $a>0,\,b>0$.
Using an appropriate change of variables, calculate the area bounded by the curves:
$$ y^2=2px,\qquad y^2=2qx,\qquad x^2=2ry,\qquad x^2=2sy, $$
where $0<p<q$ and $0<r<s$.
Using an appropriate change of variables, calculate the area bounded by the curves:
$$ x+y=a,\qquad x+y=b,\qquad y=\alpha x,\qquad y=\beta x, $$
where $a<b$ and $\alpha<\beta$.
Using an appropriate change of variables, calculate the area bounded by the curves:
$$ xy=a^2,\qquad xy=b^2,\qquad y^2=mx,\qquad y^2=nx $$
(assuming $a^2<b^2$ and $m<n$).
Using an appropriate change of variables, calculate the area bounded by the curves:
$$ xy=a^2,\qquad xy=b^2,\qquad y^2=mx,\qquad x^2=ny, $$
where $a>b$ and $m>n$.
Find the area of the region bounded by the ellipses:
$$ \frac{x^2}{\cosh^2u}+\frac{y^2}{\sinh^2u}=c^2 \qquad(u=u_1,\,u_2) $$
and the hyperbolas:
$$ \frac{x^2}{\cos^2v}-\frac{y^2}{\sin^2v}=c^2 \qquad(v=v_1,\,v_2), $$
where $0<u_1<u_2$, $0<v_1<v_2$, $x>0$, $y>0$.
Using polar coordinates, find the volume of the solid bounded by the sphere:
$$ x^2+y^2+z^2=R^2 $$
and the cylinder:
$$ x^2+y^2=Rx,\qquad x^2+y^2<Rx. $$
Using polar coordinates, find the volume of the solid bounded by the paraboloid:
$$ z=x^2+y^2, $$
the cylinders:
$$ x^2+y^2=x,\qquad x^2+y^2=2x, $$
and the plane $z=0$.
Using polar coordinates, find the volume of the solid bounded by the paraboloid of revolution:
$$ x^2+y^2-az=0, $$
the lemniscate cylinder:
$$ (x^2+y^2)^2=a^2(x^2-y^2), $$
and the plane $z=0$, where $a>0$.
Using polar coordinates, find the volume of the solid bounded by the paraboloid:
$$ 2az=x^2+y^2 $$
and the sphere:
$$ x^2+y^2+z^2=3a^2, $$
where $x^2+y^2<2az$.
Find the volume of the solid bounded by the prism whose base is the triangle with vertices:
$$ O(0,0,0),\quad A(1,0,0),\quad B\!\left(0,\frac12,0\right), $$
with edges parallel to the $z$-axis, and the surface:
$$ z=x^2+y+1 \qquad(z>0). $$
Using polar coordinates, find the volume of the solid bounded by the paraboloid:
$$ z=x^2+y^2, $$
the cylinders:
$$ x^2+y^2=x,\qquad x^2+y^2=2x, $$
and the plane $z=0$.
Find the volume of the solid bounded by the cone:
$$ z^2=a^2\big(x^2+y^2\big) $$
and the surface:
$$ x^2+y^2-bx=b\sqrt{x^2+y^2}, $$
where $a,b>0$ and $z>0$.
Find the volume of the solid bounded by the elliptic cylinder:
$$ \left(\frac{x-a}a\right)^2+\left(\frac yb\right)^2=1 $$
and the parabolic cylinder:
$$ z^2=x \qquad(a,b>0). $$
A sphere of radius $a$ centered at the origin and a circular cylinder of diameter $a$ tangent to the $yz$-plane at the origin (with axis parallel to the $z$-axis) intersect in a space curve.
1° Show that this curve is the intersection of the sphere $x^2+y^2+z^2=a^2$ with the cylinder $x^2+y^2=ax$.
2° Find the volume of the solid bounded by these two surfaces and the plane $z=0$, with $z>0$.
Find the volume of the solid bounded by the sphere:
$$ x^2+y^2+z^2=3a^2 $$
and the paraboloid:
$$ x^2+y^2=2az, $$
with $z>0$ taking the region containing the smaller of the two pieces cut off by the paraboloid (i.e. between the paraboloid and the sphere).
Find the volume of the part of the spherical shell bounded by:
$$ x^2+y^2+z^2=2 \qquad\text{and}\qquad x^2+y^2+z^2=6 $$
that lies inside the circular cylinder:
$$ x^2+z^2=x. $$
Find the volume of the solid bounded above by a surface $z$ over the region enclosed by:
$$ x^2+y^2=cx\qquad\text{and}\qquad x^4+y^4=a^2\big(x^2+y^2\big), $$
with $z=0$ below.
Find the volume of the solid in the first octant bounded by the surface:
$$ z^2=2xy $$
and the cylinder erected on the curve:
$$ \big(x^2+y^2\big)^2=2a^2xy, $$
with $x,y,z>0$.
Find the volume of the part of the sphere:
$$ x^2+y^2+z^2=a^2 $$
that satisfies:
$$ x^2+y^2\geqslant|ax|, $$
i.e. the sphere with the two symmetric off-center cylindrical regions $x^2+y^2\leqslant ax$ and $x^2+y^2\leqslant-ax$ removed.
Prove that the volume of the solid bounded by the surfaces:
$$ z=0,\qquad x^2+y^2=c^2,\qquad z\big[\varphi(x)+\varphi(y)\big]=a\varphi(x)+b\varphi(y), $$
where $\varphi$ is an arbitrary positive integrable function and $a,b>0$, is:
$$ V=\frac12\pi c^2(a+b), $$
independent of the particular choice of $\varphi$.
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the paraboloid:
$$ \frac{x^2}p+\frac{y^2}q=2z $$
and the elliptic cylinder:
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1,\qquad z=0. $$
Find the volume of the solid bounded by the surface:
$$ cz=xy, $$
the cylinder erected on the ellipse:
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, $$
and the plane $z=0$, with $x>0,\,y>0$ (and $c>0$).
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the surface:
$$ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2+\frac{z^2}{c^2}=1. $$
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the surface:
$$ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^k+\frac{z^2}{c^2}=1,\qquad z>0,\ k>0. $$
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the cone:
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{z^2}{c^2} $$
and the elliptic cylinder:
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac xh,\qquad z>0. $$
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the paraboloid:
$$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac zc $$
and the curve:
$$ \frac{x^4}{a^4}+\frac{y^4}{b^4}=\frac{x^2}{a^2}+\frac{y^2}{b^2},\qquad z=0. $$
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the surface:
$$ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2+\frac{z^4}{c^4}=1. $$
Find the volume of the solid bounded by the surface:
$$ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^n+\frac{z^{2n}}{c^{2n}}=\frac zh\left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^{n-1},\qquad n>1,\ a,b,c,h>0. $$
Find the volume of the solid bounded by the surface:
$$ \left(\frac xa+\frac yb\right)^2+\frac{z^2}{c^2}=1 $$
and the cylinder:
$$ \left(\frac xa+\frac yb\right)^2=\frac xa,\qquad y>0,\ z>0. $$
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the surface:
$$ \left[\left(\frac xa\right)^{2/3}+\left(\frac yb\right)^{2/3}\right]^3+\left(\frac zc\right)^2=1. $$
Using generalized polar coordinates $x=ar\cos^\alpha\varphi,\ y=br\sin^\alpha\varphi$ (Jacobian $J=ab\alpha r\cos^{\alpha-1}\varphi\sin^{\alpha-1}\varphi$), find the volume of the solid bounded by the surface:
$$ \left(\sqrt{\frac xa}+\sqrt{\frac yb}\right)^4+\left(\frac zc\right)^2=1,\qquad x>0,\ y>0,\ z>0. $$
Given the ellipsoid $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1$ and the paraboloid:
$$ \frac{y^2}{b^2}+\frac{z^2}{c^2}=\frac{x-\lambda}{\mu-\lambda}\cdot\frac{a^2-\mu^2}{a^2}, $$
where $-a<\lambda<\mu<a$, show that the paraboloid divides the ellipsoid into two parts, and determine $\lambda$ (in terms of $a$ and $\mu$) such that the two parts have equal volume.
Calculate the surface area of the part of the (parabolic) cylinder $z^2=4x$ that lies in the first octant, cut by the cylinder $y^2=4x$ and the plane $x=1$.
Calculate the surface area of the part of the sphere $x^2+y^2+z^2=R^2$ that projects onto the $xy$-plane outside the circle $x^2+y^2-Rx=0$, restricted to $x>0,\,y>0$ (both hemispheres).
Let $z=f(x)$, $x\in[a,b]$, be a curve in the $xz$-plane ($y=0$), with $f$ continuously differentiable. Derive the formula for the area of the surface generated by rotating this curve about the $Oz$ axis, as a double integral (and reduce it to a single integral).
Find the surface area of the part of the surface $\dfrac{x^2}a+\dfrac{y^2}b=2z$ cut out by the elliptic cylinder $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$.
Find the surface area of the part of the surface $(x^2+y^2+z^2)^2=a^2(x^2-y^2)$ that projects onto the interior of the curve $(x^2+y^2)^2=a^2(x^2-y^2)$.
1°. Show that for the surface of revolution $x=\varphi(u)\cos\theta,\ y=\varphi(u)\sin\theta,\ z=\psi(u)$, $\alpha\le u\le\beta,\ 0\le\theta<2\pi$, with $\varphi,\psi$ differentiable, the surface area is $P=2\pi\displaystyle\int_\alpha^\beta|\varphi(u)|\sqrt{\varphi'^2(u)+\psi'^2(u)}\,du$.
Find the surface areas $P_1$ and $P_2$ of the two parts cut from the ellipsoid $\dfrac{x^2+y^2}6+\dfrac{z^2}2=1$ by the cone $\dfrac{x^2+y^2}2=(z-\sqrt2)^2$.
Find the surface area of the part of the surface $z=\dfrac{x+y}{x^2+y^2}$ cut out by the cylinders $x^2+y^2=1$ and $x^2+y^2=4$, lying in the first octant.
Find the surface area of the part of the surface $z=\arctan\dfrac yx$ that projects onto the region in the plane $z=0$ bounded by the Archimedean spiral $r=\varphi$ and the axis $Ox$ (one full turn, $\varphi\in[0,2\pi]$).
Find the surface area of the part of the paraboloid $\dfrac{x^2}a+\dfrac{y^2}b=2z$ lying inside the cylinder $\left(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}\right)^2=\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}$.
Find the solid angle subtended at the origin by the rectangle $0<y<b,\,0<z<c,\,x=a>0$. Equivalently, find the area of the part of the unit sphere $x^2+y^2+z^2=1$ onto which the rectangle projects from the origin.
Find the mass of a circular annulus (ring) if at each point the surface density is inversely proportional to the square of the distance from the center of the ring.
Find the mass of a plate in the shape of an ellipse if the surface density at each point is proportional to the distance $d$ from the minor axis, and equals $\lambda$ when $d=1$.
Find the moment of inertia, with respect to the hypotenuse, of the right isosceles triangle from the previous problem: legs of length $a$, surface density $\mu(x,y)=\dfrac{k(a-x-y)}{\sqrt2}$ proportional to the distance from the hypotenuse.
A plate is bounded by the parabola $y^2=2px$ and the chord through its focus perpendicular to the axis (the latus rectum). Find the mass of the plate if the surface density at each point is inversely proportional to the distance from the directrix.
Prove that the static moment of a triangle with base $a$, with respect to that base, depends only on the height of the triangle (and not on where the apex sits horizontally).
Find the total charge $q$ distributed over the disk $x^2+y^2=ax$ if the surface charge density is $\mu=\sqrt{R^2-x^2-y^2}$ (with $R\ge a$, so that $\mu$ stays real over the whole disk).
Find the centroid of the part of the ball $x^2+y^2+z^2\le R^2$ cut out by the planes $x=a$ and $y=b$, i.e. the part with $0\le x\le a,\,0\le y\le b$ (with $a^2+b^2\le R^2$).
A sphere of radius $a$ is submerged in a liquid of density $\varrho$ to depth $h$ (measured from the center), $h>a$. Find the force of liquid pressure on the lower and upper hemispheres of the sphere's surface.
A right circular cylinder with base radius $a$ and height $h$ is fully submerged in a liquid of density $\varrho$ so that its center is at depth $h$ below the surface, and its axis makes angle $\alpha$ with the vertical. Find the force of liquid pressure on the lower and upper bases.
Find the gravitational force with which the homogeneous cylinder $x^2+y^2\le a^2,\,0\le z\le h$, of mass $M$, attracts a material point $P(0,0,b)$ ($b>h$) of mass $m$.
Prove that the moment of inertia of a circular annulus with respect to its center is twice the moment of inertia with respect to any axis through the center lying in the plane of the annulus.
Prove that the sum of the moments of inertia of a plane figure with respect to any pair of mutually perpendicular axes lying in the plane of the figure and passing through a fixed point $O$ is a constant (independent of the orientation of the pair).
Prove that the moment of inertia of a plane figure with respect to any axis equals $Md^2+I_c$, where $M$ is the total mass, $d$ is the distance from the axis to the centroid, and $I_c$ is the moment of inertia about the parallel axis through the centroid.
Prove that the volume of the solid of revolution obtained by rotating a plane figure $F$ about an axis in its plane that does not intersect $F$ equals the area $S$ of the figure multiplied by the circumference of the circle described by the centroid of $F$.