Page 188 - How To Solve Word Problems In Calculus

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Chapter 7

Problems Involving Integrals

Area Problems
If y = f (x)is nonnegative between a and b, the area of the region
bounded by its graph and the x axis between the vertical lines
x = a and x = b is the value of the definite integral

b

A = f (x) dx
a

A simple mnemonic device for remembering this formula is
to think of a rectangle, infinitesimally thin, whose height
is y = f (x) and whose width is dx. The area of this imagi-
nary rectangle would be ydx. If we “add up” all such rect-
angular areas between a and b by integrating, we obtain
b

A = ydx.
a
(x, y)
f(x)

y

a b
dx
EXAMPLE 1
Find the area of the region bounded by
2
3
y = x − 3x + 2x + 1, the x axis, and the vertical lines x = 0
and x = 2.
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