4.5 Volumes and integrals 247val -a S x 5 a or to take double the result of partitioning the interval 0 5 x 5
a. Remark: Scientists should always remember that the volume is 4ira3.
2 Supposing that 0 < h 5 2r, find the volume Y of the segment of the
spherical ball with center at the origin and radius r which lies between the planes
having the equations x = r - h and x = r. Ins.:
Y = jr f
r
h(r2 - x2) dx = rrh2(3r - h).3 The region in the first quadrant bounded by the graphs of the equations
y = kx2, x = 0, and y = fl is rotated about the y axis to produce a solid S which
is a part of a solid paraboloid like the nose of a bullet. Show that ISI, the volume
of S, is exactly half the volume of a solid circular cylinder having the same base
and altitude.
4 The region bounded by the graphs of the equations y = kx2 and y = A
is rotated about the line having the equation y = A. Find the volume of the
resulting solid. Ans.:
16af12 1115 k'
5 A region R is bounded by the graphs of the equations xy = 1, y = 0,
x = a, and x = b for which 0 < a < b. Find the volume ISI of the solid S
obtained by rotating R about the x axis. fins.: ir/a - -7r/b.
6 The region bounded by the graphs of the equations x = 1, x = 2, y = 0,
and y = Uz V9 - x2 is rotated about the x axis. Find the volume of the result-
ing solid. Ans.: 80ir/27.
7 The region bounded by the line and hyperbola having the equations
x + y = 5 and xy = 4 is rotated about the y axis. Find the volume V of the
solid thus generated. fins.: 9w.
8 Let a cylindrical shell (which resembles the part of a tomato can remaining
after the top and bottom have been removed) have length L and have inner and
outer radii rk_1 and rk. Supposing as usual that Ark = rk - rk_l, prove that
the volume of the shell is
(21rrk )L Ark,where rk* is the number defined by rk = '(rk-i + rk).
9 Set up two different integrals for the volume
of the solid torus (or anchor ring) obtained by
rotating the circular disk of Figure 4.591 about
the y axis. First make a partition of the interval
0 5 y S a of the y axis and estimate volumes of
washers (things normally associated with nuts and
bolts). Then make a partition of the interval
Figure 4.591b - a 5 x < b + a and estimate volumes of cylindrical shells (things which, if
they had tops and bottoms, would be tin cans). Evaluate one of the integrals.
Remark: The correct answer agrees with the result of applying a famous old
theorem which says that the volume of the solid of revolution is the product of
the area of the set rotated and the distance the centroid (in this case, the center)
goes. The theorem is the theorem of Pappus.