Join our free STEM summer bootcamps taught by experts. Space is limited.Register Here 🏕

Thomas Calculus

George B. Thomas Jr.

Chapter 15

Multiple Integrals

Educators

NJ
BD
+ 1 more educators

Problem 1

\begin{equation}
\begin{array}{c}{\text { a. Solve the system }} \\ {u=x-y, \quad v=2 x+y} \\ {\text { for } x \text { and } y \text { in terms of } u \text { and } v . \text { Then find the value of the }} \\ {\text { Jacobian } \partial(x, y) / \partial(u, v)}\\\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad{\text { b. Find the image under the transformation } u=x-y} \\ {v=2 x+y \text { of the triangular region with vertices }(0,0)} \\ {(1,1), \text { and }(1,-2) \text { in the } x y \text { -plane. Sketch the transformed }} \\ {\text { region in the } u v \text { -plane. }}\end{array}
\end{equation}

Check back soon!

Problem 2

a. Solve the system
$$u=x+2 y, \quad v=x-y$$
for $x$ and $y$ in terms of $u$ and $v .$ Then find the value of the Jacobian $\partial(x, y) / \partial(u, v)$
\begin{equation}
\begin{array}{l}{\text { b. Find the image under the transformation } u=x+2 y} \\ {v=x-y \text { of the triangular region in the } x y \text { -plane bounded }} \\ {\text { by the lines } y=0, y=x, \text { and } x+2 y=2 . \text { Sketch the trans- }} \\ {\text { formed region in the } u v \text { -plane. }}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 3

\begin{equation}
\begin{array}{l}{\text { a. Solve the system }} \\ {u=3 x+2 y, \quad v=x+4 y} \\ {\text { for } x \text { and } y \text { in terms of } u \text { and } v . \text { Then find the value of the }} \\ {\text { Jacobian } \partial(x, y) / \partial(u, v) .} \\ {\text { b. Find the image under the transformation } u=3 x+2 y} \\ {v=x+4 y \text { of the triangular region in the } x y \text { -plane bounded }}\\{\text { by the } x \text { -axis, the } y \text { -axis, and the line } x+y=1 . \text { Sketch the }} \\ {\text { transformed region in the } u v \text { -plane. }}\end{array}
\end{equation}

Check back soon!

Problem 4

\begin{equation}
\begin{array}{l}{\text { a. Solve the system }} \\ {u=2 x-3 y, \quad v=-x+y} \\ {\text { for } x \text { and } y \text { in terms of } u \text { and } v . \text { Then find the value of the }} \\ {\text { Jacobian } \partial(x, y) / \partial(u, v)} \\ {\text { b. Find the image under the transformation } u=2 x-3 y \text { , }} \\ {v=-x+y \text { of the parallelogram } R \text { in the } x y \text { -plane with }} \\ {\text { boundaries } x=-3, x=0, y=x, \text { and } y=x+1 . \text { Sketch }} \\ {\text { the transformed region in the } u v \text { -plane. }}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 5

Evaluate the integral
$$\int_{0}^{4} \int_{x=y / 2}^{x=(y / 2)+|} \frac{2 x-y}{2} d x d y$$
from Example 1 directly by integration with respect to $x$ and $y$ to
confirm that its value is 2 .

SG
Samuel G.
Numerade Educator

Problem 6

Use the transformation in Exercise 1 to evaluate the integral
$$\iint_{R}\left(2 x^{2}-x y-y^{2}\right) d x d y$$
for the region $R$ in the first quadrant bounded by the lines $y=-2 x+4, y=-2 x+7, y=x-2,$ and $y=x+1$

YZ
Yiming Z.
Numerade Educator

Problem 7

Use the transformation in Exercise 3 to evaluate the integral
$$\iint_{R}\left(3 x^{2}+14 x y+8 y^{2}\right) d x d y$$
for the region $R$ in the first quadrant bounded by the lines $y=-(3 / 2) x+1, y=-(3 / 2) x+3, y=-(1 / 4) x, \quad$ and $\quad y=$ $-(1 / 4) x+1 .$

Check back soon!

Problem 8

Use the transformation and parallelogram $R$ in Exercise 4 to evaluate the integral
$$\iint_{R} 2(x-y) d x d y$$

YZ
Yiming Z.
Numerade Educator

Problem 9

Let $R$ be the region in the first quadrant of the $x y$ -plane bounded
by the hyperbolas $x y=1, x y=9$ and the lines $y=x, y=4 x$ .
Use the transformation $x=u / v, y=u v$ with $u>0$ and $v>0$
to rewrite
$$\iint_{R}\left(\sqrt{\frac{y}{x}}+\sqrt{x y}\right) d x d y$$
as an integral over an appropriate region $G$ in the $u v$ -plane. Then
evaluate the $u v$ -integral over $G .$

Check back soon!

Problem 10

\begin{equation}
\begin{array}{c}\quad\quad\quad\quad{\text { a. Find the Jacobian of the transformation } x=u, y=u v \text { and }} \\ {\text { sketch the region } G : 1 \leq u \leq 2,1 \leq u v \leq 2, \text { in the }} \\ {\text { uv-plane. }} \\ {\text { b. Then use Equation }(2) \text { to transform the integral }} \\ {\int_{1}^{2} \int_{1}^{2} \frac{y}{x} d y d x}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 11

Polar moment of inertia of an elliptical plate $A$ thin plate of constant density covers the region bounded by the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1, a>0, b>0,$ in the $x y$ -plane. Find the first
moment of the plate about the origin. (Hint: Use the transformation $x=a r \cos \theta, y=b r \sin \theta . )$

Check back soon!

Problem 12

The area of an ellipse the area $\pi a b$ of the ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1$ can be found by integrating the function $f(x, y)=1$ over the region bounded by the ellipse in the $x y$ -plane.
Evaluating the integral directly requires a trigonometric substitution. An easier way to evaluate the integral is to use the transformation $x=a u, y=b v$ and evaluate the transformed integral over the
disk $G : u^{2}+v^{2} \leq 1$ in the $u v$ -plane. Find the area this way.

YZ
Yiming Z.
Numerade Educator

Problem 13

\begin{equation}
\begin{array}{c}{\text { Use the transformation in Exercise } 2 \text { to evaluate the integral }} \\ {\int_{0}^{2 / 3} \int_{y}^{2-2 y}(x+2 y) e^{(y-x)} d x d y} \\ {\text { by first writing it as an integral over a region } G \text { in the } u v \text { -plane. }}\end{array}
\end{equation}

Check back soon!

Problem 14

\begin{equation}
\begin{array}{c}{\text { Use the transformation } x=u+(1 / 2) v, y=v \text { to evaluate the }} \\ {\text { integral }} \\ {\int_{0}^{2} \int_{y / 2}^{(y+4) / 2} y^{3}(2 x-y) e^{(2 x-y)^{2}} d x d y} \\ {\text { by first writing it as an integral over a region } G \text { in the uv-plane. }}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 15

Use the transformation $x=u / v, y=u v$ to evaluate the integral sum
$$\int_{1}^{2} \int_{1 / y}^{y}\left(x^{2}+y^{2}\right) d x d y+\int_{2}^{4} \int_{y / 4}^{4 / y}\left(x^{2}+y^{2}\right) d x d y$$

Check back soon!

Problem 16

\begin{equation}
\begin{array}{c}{\text { Use the transformation } x=u^{2}-v^{2}, y=2 u v \text { to evaluate the }} \\ {\int_{0}^{1} \int_{0}^{2 \sqrt{1-x}} \sqrt{x^{2}+y^{2}} d y d x} \\ {\text { (Hint: Show that the image of the triangular region } G \text { with vertices }} \\ {(0,0),(1,0),(1,1) \text { in the } u v-\text { plane is the region of integration } R \text { in }} \\ {\text { the } x y \text { -plane defined by the limits of integration.) }}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 17

Evaluate the integral in Example 5 by integrating with respect to $x, y,$ and $z .$

Check back soon!

Problem 18

\begin{equation}
\begin{array}{c}{\text { Volume of an ellipsoid Find the volume of the ellipsoid }} \\ {\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1} \\ {\text { (Hint: Let } x=a u, y=b v, \text { and } z=c w . \text { Then find the volume of }} \\ {\text { an appropriate region in } u v w-\text { space.) }}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 19

Evaluate
\begin{equation}
\begin{array}{c} {\iiint|x y z| d x d y d z} \\ {\text { over the solid ellipsoid }} \\ {\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}} \leq 1} \\ {\text { (Hint: Let } x=a u, y=b v, \text { and } z=c w . \text { Then integrate over an }} \\ {\text { appropriate region in } u v w \text { -space.) }}\end{array}
\end{equation}

Check back soon!

Problem 20

Let $D$ be the region in $x y z$ -space defined by the inequalities
$$1 \leq x \leq 2, \quad 0 \leq x y \leq 2, \quad 0 \leq z \leq 1$$
Evaluate
$$\iiint_{D}\left(x^{2} y+3 x y z\right) d x d y d z$$
by applying the transformation
$$u=x, \quad v=x y, \quad w=3 z$$
and integrating over an appropriate region $G$ in $u v w$ -space.

YZ
Yiming Z.
Numerade Educator

Problem 21

Find the Jacobian $\partial(x, y) / \partial(u, v)$ of the transformation
\begin{equation}
\begin{array}{l}{\text { a. } x=u \cos v, \quad y=u \sin v} \\ {\text { b. } x=u \sin v, \quad y=u \cos v.}\end{array}
\end{equation}

Caleb E.
Numerade Educator

Problem 22

Find the Jacobian $\partial(x, y, z) / \partial(u, v, w)$ of the transformation
\begin{equation}
\begin{array}{l}{\text { a. } x=u \cos v, \quad y=u \sin v, \quad z=w} \\ {\text { b. } x=2 u-1, \quad y=3 v-4, \quad z=(1 / 2)(w-4)}\end{array}
\end{equation}

YZ
Yiming Z.
Numerade Educator

Problem 23

Evaluate the appropriate determinant to show that the Jacobian
of the transformation from Cartesian $\rho \phi \theta$ -space to Cartesian $x y z-$
space is $\rho^{2} \sin \phi$.

Check back soon!

Problem 24

Substitutions in single integrals How can substitutions in single definite integrals be viewed as transformations of regions? What is the Jacobian in such a case? Illustrate with an example.

Check back soon!

Problem 25

Centroid of a solid semi ellipsoid Assuming the result that the centroid of a solid hemisphere lies on the axis of symmetry three-eighths of the way from the base toward the top, show, by trans-
forming the appropriate integrals, that the center of mass of a solid semiellipsoid $\left(x^{2} / a^{2}\right)+\left(y^{2} / b^{2}\right)+\left(z^{2} / c^{2}\right) \leq 1, z \geq 0,$ lies on
the $z$ -axis three-eighths of the way from the base toward the top.
(You can do this without evaluating any of the integrals.)

Check back soon!

Problem 26

Cylindrical shells In Section $6.2,$ we learned how to find the volume of a solid of revolution using the shell method; namely, if the region between the curve $y=f(x)$ and the $x$ -axis from $a$
to $b(0 < a < b)$ is revolved about the $y$ -axis, the volume of the resulting solid is $\int_{a}^{b} 2 \pi x f(x) d x .$ Prove that finding volumes by using triple integrals gives the same result. Use cylindrical
coordinates with the roles of $y$ and $z$ changed.)

YZ
Yiming Z.
Numerade Educator

Problem 27

Inverse transform The equations $x=g(u, v), y=h(u, v)$ in Figure 15.57 transform the region $G$ in the $u v$ -plane into the region $R$ in the $x y$ -plane. Since the substitution transformation is one-
to-one with continuous first partial derivatives, it has an inverse transformation and there are equations $u=\alpha(x, y), v=\beta(x, y)$ with continuous first partial derivatives transforming $R$ back into $G .$ Moreover, the Jacobian determinants of the transformations are related reciprocally by
\begin{equation}
\frac{\partial(x, y)}{\partial(u, v)}=\left(\frac{\partial(u, v)}{\partial(x, y)}\right)^{-1}
\end{equation}
Equation $(10)$ is proved in advanced calculus. Use it to find the area of the region $R$ in the first quadrant of the $x y$ -plane bounded by the lines $y=2 x, 2 y=x,$ and the curves $x y=2,2 x y=1$ for $u=x y$ and $v=y / x .$

Check back soon!

Problem 28

(Continuation of Exercise $27 . )$ For the region $R$ described in
Exercise 27 , evaluate the integral $\iint_{R} y^{2} d A$

YZ
Yiming Z.
Numerade Educator