Calculus: Analytic Geometry and Calculus, with Vectors

248 Integrals

10 Find, in two or three different ways, the volume of the solid obtained by

replacing the disk of the preceding problem by the square with horizontal and

vertical sides tangent to the disk. One of the methods is suggested by the remark

at the end of the preceding problem.

11 Find the volume of the solid obtained by rotating, about the y axis, the

region bounded by the graphs of the equations y = 3x2 and y = 12. Ans.:

24,r.

12 Find the volume of the solid generated by rotating, about the x axis,

the region in the first quadrant bounded by the graphs of the equations y = ka,

x= 0, andy=8. Ins.:!- -

13 Let a > 0. Two circular cylinders of radius a have their axes on the

x and y axes. With axes so oriented that the z axis is vertical, sketch the part

of the first cylinder which lies in the first octant and between the planes x = 0

and x = Sa. Then sketch the part of the second cylinder which lies in the first

Figure 4.592

octant and between the planes y = 0 and y = 2a.

For three values of z, sketch the lines in which a

horizontal plane through (0,0,z) intersects the parts

of the cylinders in the figure, and then sketch the

curve in which the parts of the cylinders intersect.

If the figure is reasonably good, it should be easy

to find the volume V of the solid bounded by the

three coordinate planes and parts of the two cylin-

ders. Do it. Ins.: Figure 4.592 and V = 2a3/3.

14 Find a reason why the answer to the preced-

ing problem must be less than as.

15 A cylindrical hole is drilled through the center of a spherical ball. It is

observed that the length of the hole is L. Show that the volume of the part of the

ball remaining is the same as the volume of a spherical ball of diameter L.

16 A section of a tree trunk is a section of a right circular cylinder of radius a.

A wedge is removed by making two cuts to a diameter (line, not number), one

cut being in a horizontal plane and the other being in a plane which makes the

angle 0 with the horizontal plane. Find the volume of the wedge.

Ins.: -gal tan 0.

17 It is of interest to know that our methods are powerful enough to enable

us to derive the standard formula

(1) Y = garabc

for the volume Y of the solid in Es bounded by the ellipsoid having the equation

(2)

x2 (^2) Lz2
a2 --b2 -r i
in which a, b, c are positive constants. The formula for the volume can be
remembered with the aid of the fact that if a = b = c = r, then (2) is the (or an)
equation of a sphere of radius r and (1) gives the volume of the ball which it
bounds. To start to find the volume of the part of our solid containing points