3.7.E: Change of Variables in Definite Integrals (Exercises)

Exercise $\PageIndex{1}$

Find the area of the region enclosed by the ellipse with equation $x^{2}+4 y^{2}=4$.

Answer

$2\pi$

Exercise $\PageIndex{2}$

Given $a>0$ and $b>0$, show that the area enclosed by the ellipse with equation

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \nonumber$$

is $\pi a b$.

Exercise $\PageIndex{3}$

Find the volume of the region enclosed by the ellipsoid with equation

$$\frac{x^{2}}{25}+y^{2}+\frac{z^{2}}{4}=1 . \nonumber$$

Answer

$\frac{40 \pi}{3}$

Exercise $\PageIndex{4}$

Given $a>0$, $b>0$, and $c>0$, show that the volume of the region enclosed by the ellipsoid

$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1 \nonumber$$

is $\frac{4}{3} \pi abc$.

Exercise $\PageIndex{5}$

Find the polar coordinates for each of the following points given in Cartesian coordinates.

(a) (1,1)

(b) (-2,3)

(c) (-1,3)

(d) (4,-4)

Answer

(a) $\left(\sqrt{2}, \frac{\pi}{4}\right)$

(c) $(\sqrt{10}, 4.3906)$

Exercise $\PageIndex{6}$

Find the Cartesian coordinates for each of the following points given in polar coordinates.

(a) $(3,0)$

(b) $\left(2, \frac{5 \pi}{6}\right)$

(c) $(5, \pi)$

(d) $\left(4, \frac{4 \pi}{3}\right)$

Answer

(a) (3,0)

(c) (-5,0)

Exercise $\PageIndex{7}$

Evaluate

$$\iint_{D}\left(x^{2}+y^{2}\right) d x d y , \nonumber$$

where $D$ is the disk in $\mathbb{R}^2$ of radius 2 centered at the origin.

Answer

$\iint_{D}\left(x^{2}+y^{2}\right) d x d y=8 \pi$

Exercise $\PageIndex{8}$

Evaluate

$$\iint_{D} \sin \left(x^{2}+y^{2}\right) d x d y , \nonumber$$

where $D$ is the disk in $\mathbb{R}^2$ of radius 1 centered at the origin.

Exercise $\PageIndex{9}$

Evaluate

$$\iint_{D} \frac{1}{x^{2}+y^{2}} d x d y , \nonumber$$

where $D$ is the region in the first quadrant of $\mathbb{R}^2$ which lies between the circle with equation $x^{2}+y^{2}=1$ and the circle with equation $x^{2}+y^{2}=16$.

Answer

$\iint_{D} \frac{1}{x^{2}+y^{2}} d x d y=\pi \log (2)$

Exercise $\PageIndex{10}$

Evaluate

$$\iint_{D} \log \left(x^{2}+y^{2}\right) d x d y , \nonumber$$

where $D$ is the region in $\mathbb{R}^2$ which lies between the circle with equation $x^{2}+y^{2}=1$ and the circle with equation $x^{2}+y^{2}=4$.

Answer

$\iint_{D} \log \left(x^{2}+y^{2}\right) d x d y=\pi(8 \log (2)-3)$

Exercise $\PageIndex{11}$

Using polar coordinates, verify that the area of a circle of radius $r$ is $\pi r^2$.

Exercise $\PageIndex{12}$

Let

$$I=\int_{-\infty}^{\infty} e^{-\frac{x^{2}}{2}} d x . \nonumber$$

(a) Show that

$$I^{2}=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\frac{1}{2}\left(x^{2}+y^{2}\right)} d x d y . \nonumber$$

(b) Show that

$$I^{2}=\int_{0}^{\infty} \int_{0}^{2 \pi} r e^{-\frac{r^{2}}{2}} d \theta d r . \nonumber$$

(c) Show that

$$\int_{-\infty}^{\infty} e^{-\frac{x^{2}}{2}} d x=\sqrt{2 \pi} . \nonumber$$

Exercise $\PageIndex{13}$

Find the spherical coordinates of the point with Cartesian coordinates $(-1,1,2)$.

Answer

$\left(\sqrt{6}, \frac{3 \pi}{4}, 0.6155\right)$

Exercise $\PageIndex{14}$

Find the spherical coordinates of the point with Cartesian coordinates $(3,2,-1)$.

Exercise $\PageIndex{15}$

Find the Cartesian coordinates of the point with spherical coordinates $\left(2, \frac{3 \pi}{4}, \frac{2 \pi}{3}\right)$.

Answer

$\left(-\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}},-1\right)$

Exercise $\PageIndex{16}$

Find the Cartesian coordinates of the point with spherical coordinates $\left(5, \frac{5 \pi}{3}, \frac{\pi}{6}\right)$.

Exercise $\PageIndex{17}$

Evaluate

$$\iiint\left(x^{2}+y^{2}+z^{2}\right) d x d y d z , \nonumber$$

where $D$ is the closed ball in $\mathbb{R}^3$ of radius 2 centered at the origin.

Answer

$\iiint_{D}\left(x^{2}+y^{2}+z^{2}\right) d x d y d z=\frac{128 \pi}{5}$

Exercise $\PageIndex{18}$

Evaluate

$$\iiint_{D} \frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}} d x d y d z , \nonumber$$

where $D$ is the region in $\mathbb{R}^3$ between the two spheres with equations $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}+z^{2}=9$.

Exercise $\PageIndex{19}$

Evaluate

$$\iiint_{D} \sin \left(\sqrt{x^{2}+y^{2}+z^{2}}\right) d x d y d z ,\nonumber$$

where $D$ is the region in $\mathbb{R}^3$ described by $x \geq 0, y \geq 0, z \geq 0$, and $x^{2}+y^{2}+z^{2} \leq 1$.

Answer

$\iiint_{D} \sin \left(\sqrt{x^{2}+y^{2}+z^{2}} d x d y d z=\frac{\pi}{2}(2 \sin (1)+\cos (1)-2) \approx 0.3506\right.$

Exercise $\PageIndex{20}$

Evaluate

$$\iiint_{D} e^{-\left(x^{2}+y^{2}+z^{2}\right)} d x d y d z , \nonumber$$

where $D$ is the closed ball in $\mathbb{R}^3$ of radius 3 centered at the origin.

Exercise $\PageIndex{21}$

Let $D$ be the region in $\mathbb{R}^3$ described by $x^{2}+y^{2}+z^{2} \leq 1$ and $z \geq \sqrt{x^{2}+y^{2}}$.

(a) Explain why the spherical coordinate change of variables maps the region

$$E=\left\{(\rho, \theta, \varphi): 0 \leq \rho \leq 1,0 \leq \theta \leq 2 \pi, 0 \leq \varphi \leq \frac{\pi}{4}\right\} \nonumber$$

onto $D$.

(b) Find the volume of $D$.

Answer

(b) $\frac{\pi}{3}(2-\sqrt{2})$

Exercise $\PageIndex{22}$

If a point $P$ has Cartesian coordinates $(x,y,z)$, then the cylindrical coordinates of $P$ are $(r,\theta,z)$, where $r$ and $\theta$ are the polar coordinates of $(x,y)$. Show that

$$\left|\operatorname{det} \frac{\partial(x, y, z)}{\partial(r, \theta, z)}\right|=r . \nonumber$$

Exercise $\PageIndex{23}$

Use cylindrical coordinates to evaluate

$$\iint_{D} \sqrt{x^{2}+y^{2}} d x d y d z , \nonumber$$

where $D$ is the region in $\mathbb{R}^3$ described by $1 \leq x^{2}+y^{2} \leq 4$ and $0 \leq z \leq 5$.

Answer

$\iiint_{D} \sqrt{x^{2}+y^{2}} d x d y d z=\frac{70 \pi}{3}$

Exercise $\PageIndex{24}$

A drill with a bit with a radius of 1 centimeter is used to drill a hole through the center of a solid ball of radius 3 centimeters. What is the volume of the remaining solid?

Exercise $\PageIndex{25}$

Let $D$ be the set of all points in the intersection of the two solid cylinders in $\mathbb{R}^3$ described by $x^{2}+y^{2} \leq 1$ and $x^{2}+z^{2} \leq 1$. Find the volume of $D$.

Answer

$\frac{16}{3}$