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

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    78227
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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)

    (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

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


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