Chapter 13

For the transformation \(x=u+v, y=v-u\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=2,3,4,5\) and \(1 \leq v \leq 3)\) or \((v=1,2,3\) and \(2 \leq u \leq 5)\\}\)

In Problems 1-6, evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{2 \pi} \int_{0}^{3} \int_{0}^{12} r d z d r d \theta $$

Evaluate the iterated integrals. \(\int_{-3}^{7} \int_{0}^{2 x} \int_{y}^{x-1} d z d y d x\)

Find the area of the indicated surface. Make a sketch in each case. The part of the plane \(3 x+4 y+6 z=12\) that is above the rectangle in the \(x y\) -plane with vertices \((0,0),(2,0),(2,1)\), and \((0,1)\).

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. x=0, x=4, y=0, y=3 ; \delta(x, y)=y+1

Evaluate the iterated integrals. \(\int_{0}^{\pi / 2} \int_{0}^{\cos \theta} r^{2} \sin \theta d r d \theta\)

Evaluate each of the iterated integrals. $$ \int_{0}^{2} \int_{0}^{3}(9-x) d y d x $$

Evaluate the iterated integrals in Problems \(1-14 .\) $$ \int_{0}^{1} \int_{0}^{3 x} x^{2} d y d x $$

For the transformation \(x=2 u+v, y=v-u\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=2,3,4,5\) and \(1 \leq v \leq 3)\) or \((v=1,2,3\) and \(2 \leq u \leq 5)\\} .\)

Evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{2 \pi} \int_{1}^{3} \int_{0}^{12} r d z d r d \theta $$

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. \(y=0, y=\sqrt{4-x^{2}} ; \delta(x, y)=y\)

Evaluate the iterated integrals. \(\int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} d z d y d x\)

Find the area of the indicated surface. Make a sketch in each case. The part of the plane \(3 x-2 y+6 z=12\) that is bounded by the planes \(x=0, y=0\), and \(3 x+2 y=12\).

Evaluate the iterated integrals. \(\int_{0}^{\pi / 2} \int_{0}^{\sin \theta} r d r d \theta\)

Evaluate each of the iterated integrals. $$ \int_{-2}^{2} \int_{0}^{1}\left(9-x^{2}\right) d y d x $$

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

For the transformation \(x=u \sin v, y=u \cos v\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=0,1,2,3\) and \(0 \leq v \leq \pi)\) or \((v=0, \pi / 2, \pi\) and \(0 \leq u \leq 3)\\}\)

Evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{\pi / 4} \int_{0}^{3} \int_{0}^{9-r^{2}} z r d z d r d \theta $$

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. \(y=0, y=\sin x, 0 \leq x \leq \pi ; \delta(x, y)=y\)

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

Evaluate the iterated integrals. \(\int_{0}^{\pi} \int_{0}^{\sin \theta} r^{2} d r d \theta\)

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

Evaluate the iterated integrals. $$ \int_{-1}^{3} \int_{0}^{3 y}\left(x^{2}+y^{2}\right) d x d y $$

For the transformation \(x=u \cos v, y=u \sin v\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=0,1,2,3\) and \(0 \leq v \leq 2 \pi)\) or \((v=0, \pi, 2 \pi\) and \(0 \leq u \leq 3)\\}\)

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. \(y=1 / x, y=x, y=0, x=2 ; \delta(x, y)=x\)

Find the area of the indicated surface. Make a sketch in each case. The part of the surface \(z=\sqrt{4-y^{2}}\) in the first octant that is directly above the circle \(x^{2}+y^{2}=4\) in the \(x y\) -plane

Evaluate the iterated integrals. \(\int_{0}^{5} \int_{-2}^{4} \int_{1}^{2} 6 x y^{2} z^{3} d x d y d z\)

Evaluate the iterated integrals. \(\int_{0}^{\pi} \int_{0}^{1-\cos \theta} r \sin \theta d r d \theta\)

Evaluate each of the iterated integrals. $$ \int_{-1}^{4} \int_{1}^{2}\left(x+y^{2}\right) d y d x $$

Evaluate the iterated integrals. $$ \int_{-3}^{1} \int_{0}^{x}\left(x^{2}-y^{3}\right) d y d x $$

Evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{a} \rho^{2} \sin \phi d \rho d \theta d \phi $$

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. \(y=e^{-x}, y=0, x=0, x=1 ; \delta(x, y)=y^{2}\)

Find the area of the indicated surface. Make a sketch in each case. The part of the cylinder \(x^{2}+z^{2}=9\) that is directly over the rectangle in the \(x y\) -plane with vertices \((0,0),(2,0),(2,3)\), and \((0,3)\).

Evaluate the iterated integrals. \(\int_{4}^{24} \int_{0}^{24-x} \int_{0}^{24-x-y} \frac{y+z}{x} d z d y d x\)

Evaluate the iterated integrals. \(\int_{0}^{\pi} \int_{0}^{2} r \cos \frac{\theta}{4} d r d \theta\)

Evaluate the iterated integrals. $$ \int_{1}^{3} \int_{-y}^{2 y} x e^{y^{3}} d x d y $$

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

Evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{a} \rho^{2} \cos ^{2} \phi \sin \phi d \rho d \theta d \phi $$

For the transformation \(x=u+u /\left(u^{2}+v^{2}\right)\), \(y=v-v /\left(u^{2}+v^{2}\right)\), sketch the \(u\) -curves and \(v\) -curves for the grid \(\\{(u, v):(u=-2,-1,0,1,2\) and \(1 \leq v \leq 3)\) or \((v=1,2,3\) and \(-2 \leq u \leq 2)\\}\).

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. \(y=e^{x}, y=0, x=0, x=1 ; \delta(x, y)=2-x+y\)

Find the area of the indicated surface. Make a sketch in each case. The part of the paraboloid \(z=x^{2}+y^{2}\) that is cut off by the plane \(z=4\).

Evaluate the iterated integrals. \(\int_{0}^{5} \int_{0}^{3} \int_{z^{2}}^{9} x y z d x d z d y\)

Evaluate the iterated integrals. \(\int_{0}^{2 \pi} \int_{0}^{\theta} r d r d \theta\)

Evaluate the iterated integrals. $$ \int_{1}^{5} \int_{0}^{x} \frac{3}{x^{2}+y^{2}} d y d x $$

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

In Problems 7-14, use cylindrical coordinates to find the indicated quantity. Volume of the solid bounded by the paraboloid \(z=x^{2}+y^{2}\) and the plane \(z=4\)

Find the image of the rectangle with the given corners and find the Jacobian of the transformation. $$ x=u+2 v, y=u-2 v ;(0,0),(2,0),(2,1),(0,1) $$

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. \(r=2 \sin \theta ; \delta(r, \theta)=r\)

Find the area of the indicated surface. Make a sketch in each case. The part of the conical surface \(x^{2}+y^{2}=z^{2}\) that is directly over the triangle in the \(x y\) -plane with vertices \((0,0),(4,0)\), and \((0,4)\)

Evaluate the iterated integrals. \(\int_{0}^{2} \int_{1}^{z} \int_{0}^{\sqrt{x / z}} 2 x y z d y d x d z\)