Chapter 13

Evaluate the iterated integrals. \(\int_{0}^{\pi / 2} \int_{\sin 2 z}^{0} \int_{0}^{2 y z} \sin \left(\frac{x}{y}\right) d x d y d z\)

Find the area of the given region \(S\) by calculating \(\iint_{S} r d r d \theta .\) Be sure to make a sketch of the region first. \(S\) is the region inside the cardioid \(r=6-6 \sin \theta\).

Evaluate the iterated integrals. $$ \int_{0}^{2} \int_{-x}^{x} e^{-x^{2}} d y d x $$

Evaluate each of the iterated integrals. $$ \int_{0}^{1} \int_{0}^{1} x e^{x y} d y d x $$

Use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded above by the sphere \(r^{2}+z^{2}=5\) and below by the paraboloid \(r^{2}=4 z\)

Find the transformation from the \(u v\) -plane to the xy-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x+2 y, v=x-2 y $$

Find the moments of inertia \(I_{x}, I_{y}\), and \(I_{z}\) for the lamina bounded by the given curves and with the indicated density \(\delta .\) \(y=\sqrt{x}, x=9, y=0 ; \delta(x, y)=x+y\)

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{array}{c} S=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 3 \\ \left.0 \leq z \leq \frac{1}{6}(12-3 x-2 y)\right\\} \end{array} $$

Find the area of the given region \(S\) by calculating \(\iint_{S} r d r d \theta .\) Be sure to make a sketch of the region first. \(S\) is the region inside the larger loop of the limaçon \(r=2-4 \sin \theta\)

Evaluate the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{0}^{\sin y} e^{x} \cos y d x d y $$

Evaluate each of the iterated integrals. $$ \int_{0}^{3} \int_{0}^{1} 2 x \sqrt{x^{2}+y} d x d y $$

Use cylindrical coordinates to find the indicated quantity. Volume of the solid under the surface \(z=x y\), above the \(x y\) -plane, and within the cylinder \(x^{2}+y^{2}=2 x\)

Find the moments of inertia \(I_{x}, I_{y}\), and \(I_{z}\) for the lamina bounded by the given curves and with the indicated density \(\delta .\) y=x^{2}, y=4 ; \delta(x, y)=y

Find the area of the indicated surface. Make a sketch in each case. The part of the cylinder \(x^{2}+y^{2}=a y\) inside the sphere \(x^{2}+y^{2}+z^{2}=a^{2}, a>0 .\) Hint: Project to the \(y z\) -plane to get the region of integration.

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{array}{c} S=\left\\{(x, y, z): 0 \leq x \leq \sqrt{4-y^{2}},\right. \\ 0 \leq y \leq 2,0 \leq z \leq 3\\} \end{array} $$

Find the area of the given region \(S\) by calculating \(\iint_{S} r d r d \theta .\) Be sure to make a sketch of the region first. \(S\) is the region outside the circle \(r=2\) and inside the lemniscate \(r^{2}=9 \cos 2 \theta\).

Evaluate the iterated integrals. $$ \int_{1}^{2} \int_{0}^{x^{2}} \frac{y^{2}}{x} d y d x $$

Evaluate each of the iterated integrals. $$ \int_{0}^{1} \int_{0}^{1} \frac{y}{(x y+1)^{2}} d x d y $$

Use cylindrical coordinates to find the indicated quantity. Center of mass of the homogeneous solid bounded above by \(z=12-2 x^{2}-2 y^{2}\) and below by \(z=x^{2}+y^{2}\)

Find the transformation from the \(u v\) -plane to the xy-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}+y^{2}, v=x $$

Find the moments of inertia \(I_{x}, I_{y}\), and \(I_{z}\) for the lamina bounded by the given curves and with the indicated density \(\delta .\) Square with vertices }(0,0),(0, a),(a, a),(a, 0) ; \delta(x, y)=\\\ x+y \end{array}

Find the area of the indicated surface. Make a sketch in each case. The part of the saddle \(a z=x^{2}-y^{2}\) inside the cylinder \(x^{2}+y^{2}=a^{2}, a>0\).

Sketch the solid S. Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ S=\left\\{(x, y, z): 0 \leq x \leq \frac{1}{2} y, 0 \leq y \leq 4,0 \leq z \leq 2\right\\} $$

An iterated integral in polar coordinates is given. Sketch the region whose area is given by the iterated integral and evaluate the integral, thereby finding the area of the region. \(\int_{0}^{\pi / 4} \int_{0}^{2} r d r d \theta\)

Evaluate the iterated integrals. $$ \int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}}(x+y) d y d x $$

Evaluate each of the iterated integrals. $$ \int_{0}^{\ln 3} \int_{0}^{1} x y e^{x y^{2}} d y d x $$

Use cylindrical coordinates to find the indicated quantity. Center of mass of the homogeneous solid inside \(x^{2}+y^{2}=4\), outside \(x^{2}+y^{2}=1\), below \(z=12-x^{2}-y^{2}\), and above \(z=0\)

Find the transformation from the \(u v\) -plane to the xy-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}-y^{2}, v=x+y $$

Find the moments of inertia \(I_{x}, I_{y}\), and \(I_{z}\) for the lamina bounded by the given curves and with the indicated density \(\delta .\) Triangle with vertices \((0,0),(0, a),(a, 0) ; \delta(x, y)=\) \(x^{2}+y^{2}\)

Find the area of the indicated surface. Make a sketch in each case. The surface of the solid that is the intersection of the two solid cylinders \(x^{2}+z^{2} \leq a^{2}\) and \(x^{2}+y^{2} \leq a^{2} .\) Hint: You may need the integration formula \(\int(1+\sin \theta)^{-1} d \theta=\) \(-\tan [(\pi-2 \theta) / 4]+C\).