Chapter 13

General volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Cone Find the volume of a solid right circular cone with height \(h\) and base radius \(r\).

General volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Spherical cap Find the volume of the cap of a sphere of radius \(R\) with thickness \(h\).

Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by \(y=1+\sin x\) and \(y=1-\sin x\) on the interval \([0, \pi]\)

General volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Frustum of a cone Find the volume of a truncated solid cone of height \(h\) whose ends have radii \(r\) and \(R\).

General volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and \(h\) are positive constants. Ellipsoid Find the volume of a solid ellipsoid with axes of length \(2 a, 2 b,\) and \(2 c\).

Use double integrals to compute the area of the following regions. Make a sketch of the region. The region bounded by the lines \(x=0, x=4, y=x\), and \(y=2 x+1\)

Intersecting spheres One sphere is centered at the origin and has a radius of \(R\). Another sphere is centered at \((0,0, r)\) and has a radius of \(r,\) where \(r>R / 2 .\) What is the volume of the region common to the two spheres?

Determine whether the following statements are true and give an explanation or counterexample. a. In the iterated integral \(\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y,\) the limits \(a\) and \(b\) must be constants or functions of \(x\). b. In the iterated integral \(\int_{c}^{d} \int_{a}^{b} f(x, y) d x d y,\) the limits \(c\) and \(d\) must be functions of \(y\). c. Changing the order of integration gives \(\int_{0}^{2} \int_{1}^{y} f(x, y) d x d y=\int_{1}^{y} \int_{0}^{2} f(x, y) d y d x\).

Evaluate the following integrals. $$\iint_{R} y d A ; R=\\{(x, y): 0 \leq y \leq \sec x, 0 \leq x \leq \pi / 3\\}$$

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

Evaluate the following integrals. $$\iint_{R} \frac{x y}{1+x^{2}+y^{2}} d A ; R=\\{(x, y): 0 \leq y \leq x, 0 \leq x \leq 2\\}$$

Evaluate the following integrals. $$\iint_{R} x \sec ^{2} y d A ; R=\left\\{(x, y): 0 \leq y \leq x^{2}, 0 \leq x \leq \sqrt{\pi} / 2\right\\}$$

Draw the regions of integration and write the following integrals as a single iterated integral: $$\int_{0}^{1} \int_{e^{y}}^{e} f(x, y) d x d y+\int_{-1}^{0} \int_{e^{-y}}^{e} f(x, y) d x d y$$.

Consider the region \(R=\\{(x, y):|x|+|y| \leq 1\\}\) shown in the figure. a. Use a double integral to verify that the area of \(R\) is 2. b. Find the volume of the square column whose base is \(R\) and whose upper surface is \(z=12-3 x-4 y\). c. Find the volume of the solid above \(R\) and beneath the cylinder \(x^{2}+z^{2}=1\). d. Find the volume of the pyramid whose base is \(R\) and whose vertex is on the \(z\) -axis at (0,0,6).

Use the definition for the average value of \(a\) function over a region \(R \text { (Section } 13.1), \bar{f}=\frac{1}{\text { area of } R} \iint_{R} f(x, y) d A\). Find the average value of \(a-x-y\) over the region \(R=\\{(x, y): x+y \leq a, x \geq 0, y \geq 0\\},\) where \(a>0\).

Use the definition for the average value of \(a\) function over a region \(R \text { (Section } 13.1), \bar{f}=\frac{1}{\text { area of } R} \iint_{R} f(x, y) d A\). Find the average value of \(z=a^{2}-x^{2}-y^{2}\) over the region \(R=\left\\{(x, y): x^{2}+y^{2} \leq a^{2}\right\\},\) where \(a>0\).

Consider the following regions \(R\). a. Sketch the region \(R\). b. Evaluate \(\iint_{R} d A\) to determine the area of the region. c. Evaluate \(\iint_{R} x y d A\). \(R\) is the region between both branches of \(y=1 / x\) and the lines \(y=x+3 / 2\) and \(y=x-3 / 2\).

Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8 ). For example, under suitable conditions on \(f\), \(\int_{a}^{\infty} \int_{g(x)}^{h(x)} f(x, y) d y d x=\lim _{b \rightarrow \infty} \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) d y d x\). Use or extend the one-variable methods for improper integrals to evaluate the following integrals. $$\int_{1}^{\infty} \int_{0}^{e^{-x}} x y d y d x$$

Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8 ). For example, under suitable conditions on \(f\), \(\int_{a}^{\infty} \int_{g(x)}^{h(x)} f(x, y) d y d x=\lim _{b \rightarrow \infty} \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) d y d x\). Use or extend the one-variable methods for improper integrals to evaluate the following integrals. $$\int_{1}^{\infty} \int_{0}^{1 / x^{2}} \frac{2 y}{x} d y d x$$

Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8 ). For example, under suitable conditions on \(f\), \(\int_{a}^{\infty} \int_{g(x)}^{h(x)} f(x, y) d y d x=\lim _{b \rightarrow \infty} \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) d y d x\). Use or extend the one-variable methods for improper integrals to evaluate the following integrals. $$\int_{0}^{\infty} \int_{0}^{\infty} e^{-x-y} d y d x$$

Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 7.8 ). For example, under suitable conditions on \(f\), \(\int_{a}^{\infty} \int_{g(x)}^{h(x)} f(x, y) d y d x=\lim _{b \rightarrow \infty} \int_{a}^{b} \int_{g(x)}^{h(x)} f(x, y) d y d x\). Use or extend the one-variable methods for improper integrals to evaluate the following integrals. $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{1}{\left(x^{2}+1\right)\left(y^{2}+1\right)} d y d x$$

Compute the volume of the following solids. A tetrahedron with vertices \((0,0,0),(a, 0,0)\), \((b, c, 0),\) and \((0,0, d),\) where \(a, b, c,\) and \(d\) are positive real numbers

Compute the volume of the following solids. The column with a square base \(R=\\{(x, y):|x| \leq 1,|y| \leq 1\\}\) cut by the plane \(z=4-x-y\)

Compute the volume of the following solids. The wedge sliced from the cylinder \(x^{2}+y^{2}=1\) by the planes \(z=1-x\) and \(z=x-1\)

Compute the volume of the following solids. The wedge sliced from the cylinder \(x^{2}+y^{2}=1\) by the planes \(z=a(2-x)\) and \(z=a(x-2),\) where \(a>0\)

Let \(R_{1}=\\{(x, y): x \geq 1,1 \leq y \leq 2\\}\) and \(R_{2}=\\{(x, y): 1 \leq x \leq 2, y \geq 1\\} .\) For \(n>1,\) which integral(s) have finite values: \(\iint_{R_{1}} x^{-n} d A\) or \(\iint_{R_{2}} x^{-n} d A ?\)