Chapter 13

Find the volume of the following solids using triple integrals. The solid bounded below by the cone \(z=\sqrt{x^{2}+y^{2}}\) and bounded above by the sphere \(x^{2}+y^{2}+z^{2}=8\)

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R}(x+2 y) d A ; R=\\{(x, y): 0 \leq x \leq 3,1 \leq y \leq 4\\}$$

Find the volume of the solid below the hyperboloid \(z=5-\sqrt{1+x^{2}+y^{2}}\) and above the following regions. $$R=\\{(r, \theta): \sqrt{3} \leq r \leq 2 \sqrt{2}, 0 \leq \theta \leq 2 \pi\\}$$

Evaluate the following integrals as they are written. $$\int_{0}^{1} \int_{x}^{1} 6 y d y d x$$

Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region bounded by \(y=e^{x}, y=e^{-x}, x=0,\) and \(x=\ln 2\)

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=4 v, y=-2 u$$

Find the volume of the following solids using triple integrals. The prism in the first octant bounded by \(z=2-4 x\) and \(y=8\)

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

Find the volume of the solid below the hyperboloid \(z=5-\sqrt{1+x^{2}+y^{2}}\) and above the following regions. $$R=\\{(r, \theta): \sqrt{3} \leq r \leq \sqrt{15},-\pi / 2 \leq \theta \leq \pi\\}$$

Evaluate the following integrals as they are written. $$\int_{0}^{1} \int_{0}^{2 x} 15 x y^{2} d y d x$$

Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region bounded by \(y=\ln x,\) the \(x\) -axis, and \(x=e\)

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=2 u v, y=u^{2}-v^{2}$$

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

Find the volume of the following solids using triple integrals. The wedge above the \(x y\) -plane formed when the cylinder \(x^{2}+y^{2}=4\) is cut by the planes \(z=0\) and \(y=-z\)

Find the volume of the following solids. The solid bounded by the paraboloids \(z=x^{2}+y^{2}\) and \(z=2-x^{2}-y^{2}\)

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R} 4 x^{3} \cos y d A ; R=\\{(x, y): 1 \leq x \leq 2,0 \leq y \leq \pi / 2\\}$$

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

Find the mass and centroid (center of mass) of the following thin plates, assuming constant density. Sketch the region corresponding to the plate and indicate the location of the center of mass. Use symmetry when possible to simplify your work. The region bounded by \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=9,\) for \(y \geq 0\)

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=u \cos \pi v, y=u \sin \pi v$$

Find the volume of the following solids using triple integrals. The wedge bounded by the parabolic cylinder \(y=x^{2}\) and the planes \(z=3-y\) and \(z=0\)

Find the volume of the following solids. The solid bounded by the paraboloids \(z=2 x^{2}+y^{2}\) and \(z=27-x^{2}-2 y^{2}\)

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R} \frac{y}{\sqrt{1-x^{2}}} d A ; R=\left\\{(x, y): \frac{1}{2} \leq x \leq \frac{\sqrt{3}}{2}, 1 \leq y \leq 2\right\\}$$

Evaluate the following integrals as they are written. $$\int_{0}^{3} \int_{x^{2}}^{x+6}(x-1) d y d x$$

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=(u+v) / \sqrt{2}, y=(u-v) / \sqrt{2}$$

Evaluate the following integrals in cylindrical coordinates. $$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \int_{0}^{\sqrt{x^{2}+y^{2}}}\left(x^{2}+y^{2}\right)^{-1 / 2} d z d y d x$$

Find the volume of the following solids using triple integrals. The solid between the sphere \(x^{2}+y^{2}+z^{2}=19\) and the hyperboloid \(z^{2}-x^{2}-y^{2}=1,\) for \(z>0\)

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R} \sqrt{\frac{x}{y}} d A ; R=\\{(x, y): 0 \leq x \leq 1,1 \leq y \leq 4\\}$$

Evaluate the following integrals as they are written. $$\int_{-\pi / 4}^{\pi / 4} \int_{\sin x}^{\cos x} d y d x$$

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=u / v, y=v$$

Evaluate the following integrals in cylindrical coordinates. $$\int_{-1}^{1} \int_{0}^{1 / 2} \int_{\sqrt{3} y}^{\sqrt{1-y^{2}}}\left(x^{2}+y^{2}\right)^{1 / 2} d x d y d z$$

Find the volume of the following solids using triple integrals. The solid bounded by the surfaces \(z=e^{y}\) and \(z=1\) over the rectangle \(\\{(x, y): 0 \leq x \leq 1,0 \leq y \leq \ln 2\\}\)

Find the volume of the following solids. The solid bounded by the paraboloid \(z=8-x^{2}-3 y^{2}\) and the hyperbolic paraboloid \(z=x^{2}-y^{2}\)

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

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

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

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

Find the mass of the following objects with the given density functions. The solid cylinder \(D=\\{(r, \theta, z): 0 \leq r \leq 4,0 \leq z \leq 10\\}\) with density \(\rho(r, \theta, z)=1+z / 2\)

Find the volume of the following solids using triple integrals. The wedge of the cylinder \(x^{2}+4 y^{2}=4\) created by the planes \(z=3-x\) and \(z=x-3\)

Sketch the given region of integration \(R\) and evaluate the integral over \(R\) using polar coordinates. $$\iint_{R}\left(x^{2}+y^{2}\right) d A ; R=\\{(r, \theta): 0 \leq r \leq 4,0 \leq \theta \leq 2 \pi\\}$$

Evaluate each double integral over the region \(R\) by converting it to an iterated integral. $$\iint_{R} e^{x+2 y} d A ; R=\\{(x, y): 0 \leq x \leq \ln 2,1 \leq y \leq \ln 3\\}$$

Evaluate the following integrals as they are written. $$\int_{-2}^{2} \int_{x^{2}}^{8-x^{2}} x d y d x$$

Find the center of mass of the following plane regions with variable density. Describe the distribution of mass in the region. The upper half \((y \geq 0)\) of the disk bounded by the circle \(x^{2}+y^{2}=4\) with \(\rho(x, y)=1+y / 2\)

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

Find the mass of the following objects with the given density functions. The solid cylinder \(D=\\{(r, \theta, z): 0 \leq r \leq 3,0 \leq z \leq 2\\}\) with density \(\rho(r, \theta, z)=5 e^{-r}\)

Find the volume of the following solids using triple integrals. The solid in the first octant bounded by the cone \(z=1-\sqrt{x^{2}+y^{2}}\) and the plane \(x+y+z=1\)

Sketch the given region of integration \(R\) and evaluate the integral over \(R\) using polar coordinates. $$\iint_{R} 2 x y d A ; R=\\{(r, \theta): 1 \leq r \leq 3,0 \leq \theta \leq \pi / 2\\}$$

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

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

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 \(J(u, v)\) $$u=2 x-3 y, v=y-x$$