Chapter 13

Evaluate the following iterated integrals. $$\int_{1}^{3} \int_{0}^{\pi / 2} x \sin y d y d x$$

Sketch the following polar rectangles. $$R=\\{(r, \theta): 1 \leq r \leq 4,-\pi / 4 \leq \theta \leq 2 \pi / 3\\}$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). $$R=\\{(x, y): 0 \leq x \leq \pi / 4, \sin x \leq y \leq \cos x\\}$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=2 u v, y=u^{2}-v^{2}$$

What coordinate system is suggested if the integrand of a triple integral involves \(x^{2}+y^{2}+z^{2} ?\)

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{0}^{\ln 4} \int_{0}^{\ln 3} \int_{0}^{\ln 2} e^{-x+y+z} d x d y d z$$

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

Sketch the following polar rectangles. $$R=\\{(r, \theta): 4 \leq r \leq 5,-\pi / 3 \leq \theta \leq \pi / 2\\}$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). $$R=\left\\{(x, y): 0 \leq x \leq 2,3 x^{2} \leq y \leq-6 x+24\right\\}$$

Find the mass and center of mass of the thin rods with the following density functions. $$\rho(x)=2-x^{2} / 16, \text { for } 0 \leq x \leq 4$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=u \cos \pi v, y=u \sin \pi v$$

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq r \leq 3,0 \leq \theta \leq \pi / 3,1 \leq z \leq 4\\}$$

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{0}^{\pi / 2} \int_{0}^{1} \int_{0}^{\pi / 2} \sin \pi x \cos y \sin 2 z d y d x d z$$

Find the volume of the solid below the paraboloid \(z=4-x^{2}-y^{2}\) and above the following regions. $$R=\\{(r, \theta): 0 \leq r \leq 1,0 \leq \theta \leq 2 \pi\\}$$

Evaluate the following iterated integrals. $$\int_{1}^{4} \int_{0}^{4} \sqrt{u v} d u d v$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). $$R=\\{(x, y): 1 \leq x \leq 2, x+1 \leq y \leq 2 x+4\\}$$

Find the mass and center of mass of the thin rods with the following density functions. $$\rho(x)=2+\cos x, \text { for } 0 \leq x \leq \pi$$

Let \(S=\\{(u, v): 0 \leq u \leq 1\) \(0 \leq v \leq 1\\}\) be a unit square in the uv-plane. Find the image of \(S\) in the xy-plane under the following transformations. $$T: x=v \sin \pi u, y=v \cos \pi u$$

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq \theta \leq \pi / 2, z=1\\}$$

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\int_{0}^{2} \int_{1}^{2} \int_{0}^{1} y z e^{x} d x d z d y$$

Find the volume of the solid below the paraboloid \(z=4-x^{2}-y^{2}\) and above the following regions. $$R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq 2 \pi\\}$$

Evaluate the following iterated integrals. $$\int_{0}^{\pi / 2} \int_{0}^{1} x \cos x y d y d x$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). $$R=\left\\{(x, y): 0 \leq x \leq 4, x^{\prime} \leq y \leq 8 \sqrt{x}\right\\}$$

Find the image \(R\) in the \(x y\) -plane of the region \(S\) using the given transformation \(T\). Sketch both \(R\) and \(S\). $$S=\\{(u, v): v \leq 1-u, u \geq 0, v \geq 0\\} ; T: x=u, y=v^{2}$$

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 2 r \leq z \leq 4\\}$$

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\begin{aligned} &\iiint_{D}(x y+x z+y z) d V ; \quad D=\\{(x, y, z):-1 \leq x \leq 1\\\ &-2 \leq y \leq 2,-3 \leq z \leq 3\\} \end{aligned}$$

Find the volume of the solid below the paraboloid \(z=4-x^{2}-y^{2}\) and above the following regions. $$R=\\{(r, \theta): 1 \leq r \leq 2,0 \leq \theta \leq 2 \pi\\}$$

Evaluate the following iterated integrals. $$\int_{0}^{\ln 2} \int_{0}^{1} 6 x e^{3 y} d x d y$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). \(R\) is the triangular region with vertices \((0,0),(0,2),\) and (1,0).

Find the image \(R\) in the \(x y\) -plane of the region \(S\) using the given transformation \(T\). Sketch both \(R\) and \(S\). $$S=\left\\{(u, v): u^{2}+v^{2} \leq 1\right\\} ; T: x=2 u, y=4 v$$

Identify and sketch the following sets in cylindrical coordinates. $$\\{(r, \theta, z): 0 \leq z \leq 8-2 r\\}$$

Evaluate the following integrals. A sketch of the region of integration may be useful. $$\begin{aligned} &\iiint_{D} x y z e^{-x^{2}-y^{2}} d V ; \quad D=\\{(x, y, z): 0 \leq x \leq \sqrt{\ln 2}\\\ &0 \leq y \leq \sqrt{\ln 4}, 0 \leq z \leq 1\\} \end{aligned}$$

Find the volume of the solid below the paraboloid \(z=4-x^{2}-y^{2}\) and above the following regions. $$R=\\{(r, \theta): 1 \leq r \leq 2,-\pi / 2 \leq \theta \leq \pi / 2\\}$$

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

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). a\(R\) is the triangular region with vertices \((0,0),(0,2),\) and (1,1).

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 \(y=\sin x\) and \(y=1-\sin x\) between \(x=\pi / 4\) and \(x=3 \pi / 4\)

Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. $$\int_{0}^{2 \pi} \int_{0}^{1} \int_{-1}^{1} d z r d r d \theta$$

Find the volume of the following solids using triple integrals. The solid in the first octant bounded by the plane \(2 x+3 y+6 z=12\) and the coordinate planes

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

Evaluate the following iterated integrals. $$\int_{1}^{\ln 5} \int_{0}^{\ln 3} e^{x+y} d x d y$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). \(R\) is the region in the first quadrant bounded by a circle of radius 1 centered at the origin.

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 in the first quadrant bounded by \(x^{2}+y^{2}=16\)

Find the image \(R\) in the \(x y\) -plane of the region \(S\) using the given transformation \(T\). Sketch both \(R\) and \(S\). $$S=\\{(u, v): 2 \leq u \leq 3,3 \leq v \leq 6\\} ; T: x=u, y=v / u$$

Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. $$\int_{0}^{3} \int_{-\sqrt{9-y^{2}}}^{\sqrt{9-y^{2}}} \int_{0}^{9-3 \sqrt{x^{2}+y^{2}}} d z d x d y$$

Find the volume of the following solids using triple integrals. The solid in the first octant formed when the cylinder \(z=\sin y\) for \(0 \leq y \leq \pi,\) is sliced by the planes \(y=x\) and \(x=0\)

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

Evaluate the following iterated integrals. $$\int_{0}^{\pi / 4} \int_{0}^{3} r \sec \theta d r d \theta$$

Sketch each region and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\). \(R\) is the region in the first quadrant bounded by the \(y\) -axis and the parabolas \(y=x^{2}\) and \(y=1-x^{2}\).

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 \(y=1-|x|\) and the \(x\) -axis

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=3 u, y=-3 v$$