Page 259 - Calculus Demystified

CHAPTER 8
246
EXAMPLE 8.24 Applications of the Integral
Find the surface area of a right circular cone with base of radius 4 and
height 8.
SOLUTION
It is convenient to think of such a cone as the surface obtained by rotating
the graph of f(x) = x/2, 0 ≤ x ≤ 8, about the x-axis (Fig. 8.34). According
to our definition, the surface area of the cone is
8 x 5 8
√
2 1/2
2π [1 + (1/2) ] dx = 2π xdx
0 2 4 0
√
= 16 5π.

y

x

Fig. 8.34

You Try It: The standard formula for the surface area of a cone is

2
2
S = πr h + r .
Derive this formula by the method of Example 8.24.
We may also consider the area of a surface obtained by rotating the graph of a
function about the y-axis. We do so by using y as the independent variable. Here
is an example:
EXAMPLE 8.25
Set up, but do not evaluate, the integral for finding the area of the surface
6
obtained when the graph of f(x) = x , 1 ≤ x ≤ 4, isrotated about the
y-axis.