9. Graphical Applications of Integrals

Use calculus to find the volume of a tetrahedron (pyramid with four triangular faces), all of whose edges have length 4.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=1 / 4√1 − x^2,y=0,x=0, and x=12; about the x-axis

Find the volume of the solid obtained by rotating the region bounded by $y=x+4$, $y=0$, $x=1$ & $x=5$ about the x-axis.

Volumes

Volume of a solid sphere hole A round hole of radius √3 ft is bored through the center of a solid sphere of radius 2 ft. Find the volume of material removed from the sphere.

Comparing volumes Let R be the region bounded by y=1/x^p and the x-axis on the interval [1, a], where p>0 and a>1 (see figure). Let Vₓ and Vᵧ be the volumes of the solids generated when R is revolved about the x- and y-axes, respectively.

d. Find a general expression for Vᵧ in terms of a and p. Note that p=2 is a special case. What is Vᵧ when p=2?

Volumes

Find the volumes of the solids in Exercises 1–18.

The solid lies between planes perpendicular to the x-axis at x = 0 and x = 4. The cross-sections of the solid perpendicular to the x-axis between these planes are circular disks whose diameters run from the curve x² = 4y to the curve y² = 4x.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=|x| and y=2−x^2; about the x-axis 

6–8. Let R be the region bounded by the curves y = 2−√x,y=2, and x=4 in the first quadrant.

Suppose the shell method is used to determine the volume of the solid generated by revolving R about the line x=4.

a. What is the radius of a cylindrical shell at a point x in [0, 4]?

Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures). 

c. Write an integral for the volume of the solid.

64–68. Shell method Use the shell method to find the volume of the following solids.

A right circular cone of radius 3 and height 8

For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.

R is bounded by y=4−2x, the x-axis, and the y-axis.

Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

b. Repeat part (a) using the disk method.

120. Equal volumes

a. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?

b. Repeat part (a) on the interval [0, 1].

119. {Use of Tech} Comparing volumes Let R be the region bounded by y = ln(x), the x-axis, and the line x = a, where a > 1.

b. Find the volume V₂(a) of the solid generated when R is revolved about the y-axis (as a function of a).