Chapter 15

\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}

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ r=2 $$

Finding a center of mass Find the center of mass of a thin plate of density \(\delta=3\) bounded by the lines \(x=0, y=x,\) and the parabola \(y=2-x^{2}\) in the first quadrant.

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The coordinate axes and the line \(x + y = 2\)

Sketch the described regions of integration. \begin{equation}0 \leq x \leq 3, \quad 0 \leq y \leq 2 x\end{equation}

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{1}^{2} \int_{0}^{4} 2 x y d y d x$$

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}

Volume of rectangular solid integrals for the volume of the rectangular solid in the first octant bounded by the coordinate planes and the planes \(x=1, y=2,\) and \(z=3 .\) Evaluate one of the integrals.

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The lines \(x = 0 , y = 2 x ,\) and \(y = 4\)

Sketch the described regions of integration. \begin{equation}-1 \leq x \leq 2, \quad x-1 \leq y \leq x^{2}\end{equation}

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{0}^{2} \int_{-1}^{1}(x-y) d y d x$$

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ \theta=\frac{\pi}{4} $$

\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}

Volume of tetrahedron Write six different iterated triple integrals for the volume of the tetrahedron cut from the first octant by the plane \(6 x+3 y+2 z=6 .\) Evaluate one of the integrals.

Finding a centroid Find the centroid of the region in the first quadrant bounded by the \(x\) -axis, the parabola \(y^{2}=2 x,\) and the line \(x+y=4 .\)

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabola \(x = - y ^ { 2 }\) and the line \(y = x + 2\)

Sketch the described regions of integration. \begin{equation}-2 \leq y \leq 2, \quad y^{2} \leq x \leq 4\end{equation}

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{-1}^{0} \int_{-1}^{1}(x+y+1) d x d y$$

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ z=-1 $$

\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}

Volume of solid Write six different iterated triple integrals for the volume of the region in the first octant enclosed by the cylinder \(x^{2}+z^{2}=4\) and the plane \(y=3 .\) Evaluate one of the integrals.

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ z=r $$

Finding a centroid Find the centroid of the triangular region cut from the first quadrant by the line \(x+y=3 .\)

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabola \(x = y - y ^ { 2 }\) and the line \(y = - x\)

Sketch the described regions of integration. \begin{equation}0 \leq y \leq 1, \quad y \leq x \leq 2 y\end{equation}

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{0}^{1} \int_{0}^{1}\left(1-\frac{x^{2}+y^{2}}{2}\right) d x d y$$

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 .

Volume enclosed by paraboloids Let\(D\) be the region bounded by the paraboloids \(z=8-x^{2}-y^{2}\) and \(z=x^{2}+y^{2}\) . Write six different triple iterated integrals for the volume of \(D .\) Evaluate one of the integrals.

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ r=\theta $$

Finding a centroid Find the centroid of the region cut from the first quadrant by the circle \(x^{2}+y^{2}=a^{2}\) .

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The curve \(y = e ^ { x }\) and the lines \(y = 0 , x = 0 ,\) and \(x = \ln 2\)

Sketch the described regions of integration. \begin{equation}0 \leq x \leq 1, \quad e^{x} \leq y \leq e\end{equation}

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{0}^{3} \int_{0}^{2}\left(4-y^{2}\right) d y d x$$

Volume inside paraboloid beneath a plane Let \(D\) be the region bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=2 y\) . Write triple iterated integrals in the order \(d z d x d y\) and \(d z d y d x\) that give the volume of \(D .\) Do not evaluate either integral.

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ z=r \sin \theta $$

Finding a centroid Find the centroid of the region between the \(x\) -axis and the arch \(y=\sin x, 0 \leq x \leq \pi\)

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The curves \(y = \ln x\) and \(y = 2 \ln x\) and the line \(x = e ,\) in the first quadrant

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{0}^{3} \int_{-2}^{0}\left(x^{2} y-2 x y\right) d y d x$$

Sketch the described regions of integration. \begin{equation}1 \leq x \leq e^{2}, \quad 0 \leq y \leq \ln x\end{equation}

Evaluate the integrals in Exercises \(7-20\) $$ \int_{0}^{1} \int_{0}^{1} \int_{0}^{1}\left(x^{2}+y^{2}+z^{2}\right) d z d y d x $$

In Exercises \(1-8,\) describe the given region in polar coordinates. The region enclosed by the circle \(x^{2}+y^{2}=2 x\)

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ r^{2}+z^{2}=4 $$

Finding moments of inertia Find the moment of inertia about the \(x\) -axis of a thin plate of density \(\delta=1 \mathrm{gm} / \mathrm{cm}^{2}\) bounded by the circle \(x^{2}+y^{2}=4 .\) Then use your result to find \(I_{y}\) and \(I_{0}\) for the plate.

In Exercises \(1 - 12 ,\) sketch the region bounded by the given lines and curves. Then express the region's area as an iterated double integral and evaluate the integral. The parabolas \(x = y ^ { 2 }\) and \(x = 2 y - y ^ { 2 }\)

In Exercises \(1-14,\) evaluate the iterated integral. $$\int_{0}^{1} \int_{0}^{1} \frac{y}{1+x y} d x d y$$

Sketch the described regions of integration. \begin{equation}0 \leq y \leq 1, \quad 0 \leq x \leq \sin ^{-1} y\end{equation}

Evaluate the integrals in Exercises \(7-20\) $$ \int_{0}^{\sqrt{2}} \int_{0}^{3 y} \int_{x^{2}+3 y^{2}}^{8-x^{2}-y^{2}} d z d x d y $$

In Exercises \(1-8,\) describe the given region in polar coordinates. The region enclosed by the semicircle \(x^{2}+y^{2}=2 y, y \geq 0\)

Finding a moment of inertia Find the moment of inertia with respect to the \(y\) -axis of a thin sheet of constant density \(\delta=1 \mathrm{gm} / \mathrm{cm}^{2}\) bounded by the curve \(y=\left(\sin ^{2} x\right) / x^{2}\) and the interval \(\pi \leq x \leq 2 \pi\) of the \(x\) -axis.

In Exercises \(1-12,\) sketch the graph described by the following cylindrical coordinates in three-dimensional space. $$ 1 \leq r \leq 2, \quad 0 \leq \theta \leq \frac{\pi}{3} $$