Page 193 - How To Solve Word Problems In Calculus

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If the curves intersect at more than two locations, the area
must be computed by subdividing the interval, integrating
separately in each subinterval, and adding the absolute values
of the integrals, in a manner similar to Example 3.
To extend our mnemonic device to areas bounded by two
curves, consider an infinitesimally thin rectangle of width dx
extending from y 1 = g(x)to y 2 = f (x). Its height is y 2 − y 1
and its area is (y 2 − y 1 )dx. The area of the region, obtained
b

by integrating (adding), becomes (y 2 − y 1 ) dx.
a
f(x)
(x, y )
2

a dx b
g(x)
(x, y )
1
EXAMPLE 4
Determine the area of the region bounded by the parabola
2
y = 9 − x and the line x + y = 7.
Solution
The parabola is represented by the function
2
y 2 = f (x) = 9 − x . To determine g(x) we solve the line’s equa-
tion for y:
x + y = 7

y = 7 − x
y 1 = g(x) = 7 − x

We will need the points of intersection of these two curves.
This is accomplished by solving the equation f (x) = g(x) for x.

2
9 − x = 7 − x
2
0 = x − x − 2
0 = (x + 1)(x − 2)

x =−1 x = 2

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