Chapter 13

In Problems 1-10, find the mass \(m\) and center of mass \((\bar{x}, \bar{y})\) of the lamina bounded by the given curves and with the indicated density. \(r=2+2 \cos \theta ; \delta(r, \theta)=r\)

In Problems 1–10, evaluate the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{\sin 2 z}^{0} \int_{0}^{2 y z} \sin \left(\frac{x}{y}\right) d x d y d z $$

In Problems \(7-14\), 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 \(r^{2}=4 z\)

Evaluate the iterated integrals in Problems 1-14. \(\int_{0}^{\pi / 2} \int_{0}^{\sin y} e^{x} \cos y d x d y\)

In Problems \(11-16\), find the transformation from the uv-plane to the \(x y\)-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x+2 y, v=x-2 y $$

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

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{gathered} S=\\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 3 \\ \left.0 \leq z \leq \frac{1}{6}(12-3 x-2 y)\right\\} \end{gathered} $$

\(S\) is the region outside the circle \(r=2\) and inside the lemniscate \(r^{2}=9 \cos 2 \theta\).

In Problems \(7-14\), 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 \(x^{2}+y^{2}=2 x\)

Evaluate the iterated integrals in Problems 1-14. \(\int_{1}^{2} \int_{0}^{x^{2}} \frac{y^{2}}{x} d y d x\)

In Problems \(11-16\), find the transformation from the uv-plane to the \(x y\)-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=2 x-3 y, v=3 x-2 y $$

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

In Problems 11-14, 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 \(\delta\). \(y=x^{2}, y=4 ; \delta(x, y)=y\)

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{gathered} S=\left\\{(x, y, z): 0 \leq x \leq \sqrt{4-y^{2}}\right. \\ 0 \leq y \leq 2,0 \leq z \leq 3\\} \end{gathered} $$

In Problems 13-18, 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. $$ \int_{0}^{\pi / 4} \int_{0}^{2} r d r d \theta $$

In Problems \(7-14\), 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 \(z=x^{2}+y^{2}\)

Evaluate the iterated integrals in Problems 1-14. \(\int_{0}^{2} \int_{0}^{\sqrt{4-x^{2}}}(x+y) d y d x\)

In Problems \(11-16\), find the transformation from the uv-plane to the \(x y\)-plane and find the Jacobian. Assume that \(x \geq 0\) and \(y \geq 0\). $$ u=x^{2}+y^{2}, v=x $$

The partition of \(R\) into six equal squares by the lines \(x=2, x=4\), and \(y=2\). Approximate \(\iint_{R} f(x, y) d A\) by calculating the corresponding Riemann sum \(\sum_{k=1}^{6} f\left(\bar{x}_{k}, \bar{y}_{k}\right) \Delta A_{k}\), assuming that \(\left(\bar{x}_{k}, \bar{y}_{k}\right)\) are the centers of the six squares. \(f(x, y)=\sqrt{x+y}\)

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