Page 235 - Calculus Demystified

CHAPTER 8
222
EXAMPLE 8.3 Applications of the Integral
A solid has base in the x-y plane consisting of a unit square with center at
the origin and vertices on the axes. The vertical cross-section at position x
isitself a square. Refer to Fig. 8.9. What isthe volume of thissolid?

Fig. 8.9

SOLUTION
It is sufficient to calculate the volume of the right half of this solid, and to
√
double the answer. Of course the extent of x is then 0 ≤ x ≤ 1/ 2.At position
√
x, the height of the upper edge of the square base is 1/ 2 − x. So the base of
√
the vertical square slice is 2(1/ 2−x) (Fig. 8.10). The area of the slice is then

√ 2 √ 2
A(x) = 2 1/ 2 − x = 2 − 2x .

_
2(1/√2 x)

_
2(1/√2 x)
Fig. 8.10

It follows that
√
1/ 2
V = 2 · A(x) dx
0
√

1/ 2 √ 2
= 2 2 − 2x dx
0