Chapter 13

Find the mass of the following objects with the given density functions. The solid cone \(D=\\{(r, \theta, z): 0 \leq z \leq 6-r, 0 \leq r \leq 6\\}\) with density \(\rho(r, \theta, z)=7-z\)

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

Sketch the given region of integration \(R\) and evaluate the integral over \(R\) using polar coordinates. $$\iint_{R} 2 x y d A ; R=\left\\{(x, y): x^{2}+y^{2} \leq 9, y \geq 0\right\\}$$

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R}\left(x^{5}-y^{5}\right)^{2} d A ; R=\\{(x, y): 0 \leq x \leq 1,-1 \leq y \leq 1\\}$$

Evaluate the following integrals as they are written. $$\int_{0}^{1} \int_{0}^{x} 2 e^{x^{2}} d y d x$$

Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. The quarter disk in the first quadrant bounded by \(x^{2}+y^{2}=4\) with \(\rho(x, y)=1+x^{2}+y^{2}\)

Solve the following relations for \(x\) and \(y,\) and compute the Jacobian \(J(u, v)\) $$u=x+4 y, v=3 x+2 y$$

Find the mass of the following objects with the given density functions. The solid paraboloid \(D=\left\\{(r, \theta, z): 0 \leq z \leq 9-r^{2}\right.\) \(0 \leq r \leq 3\\}\) with density \(\rho(r, \theta, z)=1+z / 9\)

Sketch the given region of integration \(R\) and evaluate the integral over \(R\) using polar coordinates. $$\iint_{R} \frac{d A}{1+x^{2}+y^{2}} ; R=\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq \pi\\}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} y \cos x y d A ; R=\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq \pi / 3\\}$$

Evaluate the following integrals as they are written. $$\int_{0}^{\sqrt[3]{\pi / 2}} \int_{0}^{x} y \cos x^{3} d y d x$$

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The upper half of the ball \(x^{2}+y^{2}+z^{2} \leq 16(\text { for } z \geq 0)\)

Which weighs more? For \(0 \leq r \leq 1,\) the solid bounded by the cone \(z=4-4 r\) and the solid bounded by the paraboloid \(z=4-4 r^{2}\) have the same base in the \(x y\) -plane and the same height. Which object has the greater mass if the density of both objects is \(\rho(r, \theta, z)=10-2 z ?\)

Sketch the given region of integration \(R\) and evaluate the integral over \(R\) using polar coordinates. $$\begin{aligned} &\iint_{R} \frac{d A}{\sqrt{16-x^{2}-y^{2}}}\\\ &R=\left\\{(x, y): x^{2}+y^{2} \leq 4, x \geq 0, y \geq 0\right\\} \end{aligned}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R}(y+1) e^{x(y+1)} d A ; R=\\{(x, y): 0 \leq x \leq 1,-1 \leq y \leq 1\\}$$

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} x y d A ; R\) is bounded by \(x=0, y=2 x+1,\) and \(y=-2 x+5\).

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The upper half of the ball \(x^{2}+y^{2}+z^{2} \leq 16(\text { for } z \geq 0)\)

To evaluate the following integrals, carry out these steps. a. Sketch the original region of integration \(R\) in the xy-plane and the new region \(S\) in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to \(u\) and \(v\) c. Compute the Jacobian. d. Change variables and evaluate the new integral. \(\iint_{R} x^{2} y d A,\) where \(R=\\{(x, y): 0 \leq x \leq 2, x \leq y \leq x+4\\}\) use \(x=2 u, y=4 v+2 u\).

Evaluate the following integrals. $$\int_{1}^{6} \int_{0}^{4-2 y / 3} \int_{0}^{12-2 y-3 z} \frac{1}{y} d x d z d y$$

Sketch the given region of integration \(R\) and evaluate the integral over \(R\) using polar coordinates. $$\iint_{R} e^{-x^{2}-y^{2}} d A ; R=\left\\{(x, y): x^{2}+y^{2} \leq 9\right\\}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} x \sec ^{2} x y d A ; R=\\{(x, y): 0 \leq x \leq \pi / 3,0 \leq y \leq 1\\}$$

Evaluate the following integrals. A sketch is helpful. \(\iint_{R}(x+y) d A ; R\) is the region in the first quadrant bounded by \(x=0, y=x^{2},\) and \(y=8-x^{2}\).

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The tetrahedron in the first octant bounded by \(z=1-x-y\) and the coordinate planes

Use cylindrical coordinates to find the volume of the following solids. The solid bounded by the plane \(z=0\) and the hyperboloid \(z=\sqrt{17}-\sqrt{1+x^{2}+y^{2}}\)

Evaluate the following integrals. $$\int_{0}^{3} \int_{0}^{\sqrt{9-z^{2}}} \int_{0}^{\sqrt{1+x^{2}+z^{2}}} d y d x d z$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} 6 x^{5} e^{x^{3} y} d A ; R=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 2\\}$$

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} y^{2} d A ; R\) is bounded by \(x=1, y=2 x+2,\) and \(y=-x-1\).

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The solid bounded by the cone \(z=16-r\) and the plane \(z=0\)

To evaluate the following integrals, carry out these steps. a. Sketch the original region of integration \(R\) in the xy-plane and the new region \(S\) in the uv-plane using the given change of variables. b. Find the limits of integration for the new integral with respect to \(u\) and \(v\) c. Compute the Jacobian. d. Change variables and evaluate the new integral. \(\iint_{R} x y d A,\) where \(R\) is bounded by the ellipse \(9 x^{2}+4 y^{2}=36\) use \(x=2 u, y=3 v\).

Use cylindrical coordinates to find the volume of the following solids. The solid bounded by the plane \(z=25\) and the paraboloid \(z=x^{2}+y^{2}\)

Evaluate the following integrals. $$\int_{0}^{\pi} \int_{0}^{\pi} \int_{0}^{\sin x} \sin y d z d x d y$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} y^{3} \sin x y^{2} d A ; R=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq \sqrt{\pi / 2}\\}$$

Evaluate the following integrals. A sketch is helpful. \(\iint_{R} x^{2} y d A ; R\) is the region in quadrants 1 and 4 bounded by the semicircle of radius 4 centered at (0, 0).

Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, \(R\) and \(S\) $$\int_{0}^{1} \int_{y}^{y+2} \sqrt{x-y} d x d y$$

Use cylindrical coordinates to find the volume of the following solids. The solid bounded by the plane \(z=\sqrt{29}\) and the hyperboloid \(z=\sqrt{4+x^{2}+y^{2}}\)

Evaluate the following integrals. $$\int_{1}^{\ln 8} \int_{1}^{\sqrt{z}} \int_{\ln y}^{\ln 2 y} e^{x+y^{2}-z} d x d y d z$$

The surface of an island is defined by the following functions over the region on which the function is non-negative. Find the volume of the island. $$z=25-\sqrt{x^{2}+y^{2}}$$

When converted to an iterated integral, the following double integrals are easier to evaluate in one order than the other. Find the best order and evaluate the integral. $$\iint_{R} \frac{x}{(1+x y)^{2}} d A ; R=\\{(x, y): 0 \leq x \leq 4,1 \leq y \leq 2\\}$$

Find the center of mass of the following solids, assuming a constant density of 1. Sketch the region and indicate the location of the centroid. Use symmetry when possible and choose a convenient coordinate system. The solid bounded by the upper half \((z \geq 0)\) of the ellipsoid \(4 x^{2}+4 y^{2}+z^{2}=16\)

Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration, \(R\) and \(S\) $$\begin{aligned} &\iint_{R} \sqrt{y^{2}-x^{2}} d A, \text { where } R \text { is the diamond bounded by }\\\ &y-x=0, y-x=2, y+x=0, \text { and } y+x=2 \end{aligned}$$

Use cylindrical coordinates to find the volume of the following solids. The solid cylinder whose height is 4 and whose base is the disk \(\\{(r, \theta): 0 \leq r \leq 2 \cos \theta\\}\)

Evaluate the following integrals. $$\int_{0}^{1} \int_{0}^{\sqrt{1-x^{2}}} \int_{0}^{2-x} 4 y z d z d y d x$$

The surface of an island is defined by the following functions over the region on which the function is non-negative. Find the volume of the island. $$z=\frac{20}{1+x^{2}+y^{2}}-2$$

Average value Compute the average value of the following functions over the region \(R\). $$f(x, y)=4-x-y ; R=\\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 2\\}$$

Sketch the following regions \(R\). Then express \(\iint_{R} f(r, \theta) d A\) as an iterated integral over \(R\). The region inside the limaçon \(r=1+\frac{1}{2} \cos \theta\)

Evaluate the following integrals. $$\int_{0}^{2} \int_{0}^{4} \int_{y^{2}}^{4} \sqrt{x} d z d x d y$$

Average value Compute the average value of the following functions over the region \(R\). $$f(x, y)=e^{-y} ; R=\\{(x, y): 0 \leq x \leq 6,0 \leq y \leq \ln 2\\}$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d x d y\). The region bounded by \(y=2 x+3, y=3 x-7,\) and \(y=0\)

Use cylindrical coordinates to find the volume of the following solids. The solid in the first octant bounded by the cylinder \(r=1,\) and the planes \(z=x\) and \(z=0\)

Find the coordinates of the center of mass of the following solids with variable density. The solid bounded by the paraboloid \(z=4-x^{2}-y^{2}\) and \(z=0\) with \(\rho(x, y, z)=5-z\)