3.7.E: Change of Variables in Definite Integrals (Exercises)
- Page ID
- 78227
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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
-
\(\frac{16}{3}\)