APPLICAnONS 8.2 TO GEOMETRY 435
14. Rotate the bell-shaped curve y = e-xz/2 shown in Figure 8.21 around the y-axis, forming a
hill-shaped solid of revolution. By slicing horizontally, find the volume of this hill.
y
1
Y = e-xzJ2
x
Figure 8.21
15. The circumference of the trunk of a certain tree at different heights above the ground is given
in the following table.
Height (feet) 120"
Circumference (feet) 1
.
I
Assume all horizontal cross-sections of the trunk are circles. Estimate the volume of the tree
trunk using the trapezoid rule.
16. Most states expect to run out of space for their garbage soon. In New York, solid garbage is
packed into pyramid-shaped dumps with square bases. (The largest such dump is on Staten
Island.) A small community has a dump with a base length of 100 yards. One yard vertically
above the base, the length of the side parallel to the base is 99 yards; the dump can be built up
to a vertical height of 20 yards. (The top of the pyramid is never reached.) 65 If cubic yards
of garbage arrive at the dump every day, how long will it be before the dump is full?
17. The hull of a certain boat has widths given by the following table. Reading across a row of the
table gives widths at points 0, 10, . . . , 60 feet from the front to the back at a certain level below
waterline. Reading down a column of the table gives widths at levels 0, 2, 4, 6, 8 feet below
waterline at a certain point from the front to the back. Use the trapezoidal rule to estimate the
volume of the hull below waterline.
Front of boat ---+ Back of boat
0 10 20 30 40 50 60
0 2 8 13 16 17 16 10
Depth 2 1 4 8 10 11 10 8
below 4 0 3 4 6 7 6 4
waterline 6 0 1 2 3 4 3 2
(in feet) 8 0 0 1 1 1 1 1
For Problems 18 and 19, find the arc length of the given function from x = 0 to x = 2.
18. f(x) = ~ 19. f(x) = R
20. (a) Write an integral which represents the circumference of a circle of radius r.
(b) Evaluate the integral, and show that you get the answer you expect.
.