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

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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)

(d) (4,-4)

Answer

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

Exercise \(\PageIndex{6}\)

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

(a) \((3,0)\)

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

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

Answer

(a) (3,0)

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

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

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

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: \(\frac{16}{3}\)