Chapter 13

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{0}^{\cos \theta} r^{2} \sin \theta d r d \theta $$

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 in Problems 1-14. \(\int_{0}^{1} \int_{0}^{3 x} x^{2} d y d x\)

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 \(2 \leq u \leq 5)\\}\).

Let \(R=\\{(x, y): 1 \leq x \leq 4,0 \leq y \leq 2\\}\). Evaluate \(\iint_{R} f(x, y) d A\), where \(f\) is the given function. \(f(x, y)= \begin{cases}2 & 1 \leq x<3,0 \leq y \leq 2 \\ 3 & 3 \leq x \leq 4,0 \leq y \leq 2\end{cases}\)

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

In Problems \(1-17\), find the area of the indicated surface. Make a sketch in each case. 1\. 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 \((0,1)\)

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

In Problems 1–10, evaluate the iterated integrals. $$ \int_{-3}^{7} \int_{0}^{2 x} \int_{y}^{x-1} d z d y d x $$

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{\pi / 2} \int_{0}^{\sin \theta} r d r d \theta $$

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

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

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 \(2 \leq u \leq 5)\\}\).

Let \(R=\\{(x, y): 1 \leq x \leq 4,0 \leq y \leq 2\\}\). Evaluate \(\iint_{R} f(x, y) d A\), where \(f\) is the given function. \(f(x, y)=\left\\{\begin{array}{rl}-1 & 1 \leq x \leq 4,0 \leq y<1 \\ 2 & 1 \leq x \leq 4,1 \leq y \leq 2\end{array}\right.\)

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

The part of the plane \(3 x-2 y+6 z=12\) that is bounded by the planes \(x=0, y=0\), and \(3 x+2 y=12\)

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

In Problems 1–10, evaluate the iterated integrals. $$ \int_{0}^{2} \int_{-1}^{4} \int_{0}^{3 y+x} d z d y d x $$

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{\pi} \int_{0}^{\sin \theta} r^{2} d r d \theta $$

In Problems 1-6, evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. \(\int_{0}^{\pi / 4} \int_{0}^{3} \int_{0}^{9-r^{2}} z r d z d r d \theta\)

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

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 \(0 \leq u \leq 3)\\}\).

Let \(R=\\{(x, y): 1 \leq x \leq 4,0 \leq y \leq 2\\}\). Evaluate \(\iint_{R} f(x, y) d A\), where \(f\) is the given function. \(f(x, y)= \begin{cases}2 & 1 \leq x<3,0 \leq y<1 \\ 1 & 1 \leq x<3,1 \leq y \leq 2 \\ 3 & 3 \leq x \leq 4,0 \leq y \leq 2\end{cases}\)

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

The part of the surface \(z=\sqrt{4-y^{2}}\) that is directly above the square in the \(x y\)-plane with vertices \((1,0),(2,0),(2,1)\), and \((1,1)\)

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

In Problems 1–10, evaluate the iterated integrals. $$ \int_{1}^{4} \int_{z-1}^{2 z} \int_{0}^{y+2 z} d x d y d z $$

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{\pi} \int_{0}^{1-\cos \theta} r \sin \theta d r d \theta $$

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

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

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 \(0 \leq u \leq 3)\\}\).

Let \(R=\\{(x, y): 1 \leq x \leq 4,0 \leq y \leq 2\\}\). Evaluate \(\iint_{R} f(x, y) d A\), where \(f\) is the given function. \(f(x, y)= \begin{cases}2 & 1 \leq x \leq 4,0 \leq y<1 \\ 3 & 1 \leq x<3,1 \leq y \leq 2 \\ 1 & 3 \leq x \leq 4,1 \leq y \leq 2\end{cases}\)

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

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

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

In Problems 1–10, evaluate the iterated integrals. $$ \int_{0}^{5} \int_{-2}^{4} \int_{1}^{2} 6 x y^{2} z^{3} d x d y d z $$

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{\pi} \int_{0}^{2} r \cos \frac{\theta}{4} d r d \theta $$

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

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

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

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

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 \((0,3)\)

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

In Problems 1–10, evaluate the iterated integrals. $$ \int_{4}^{24} \int_{0}^{24-x} \int_{0}^{24-x-y} \frac{y+z}{x} d z d y d x $$

In Problems 1-6, evaluate the iterated integrals. $$ \int_{0}^{2 \pi} \int_{0}^{\theta} r d r d \theta $$

In Problems 1-6, evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. \(\int_{0}^{\pi / 2} \int_{0}^{\pi / 2} \int_{0}^{a} \rho^{2} \cos ^{2} \phi \sin \phi d \rho d \theta d \phi\)

Evaluate the iterated integrals in Problems 1-14. \(\int_{1}^{5} \int_{0}^{x} \frac{3}{x^{2}+y^{2}} d y d x\)

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 \(-2 \leq u \leq 2)\\}\).

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

The part of the paraboloid \(z=x^{2}+y^{2}\) that is cut off by the plane \(z=4\)