Page 128 - Calculus Demystified

The Integral
CHAPTER 4
discussion preceding this example, we know then that 115
1 4

Area = f(x) dx − f(x) dx
−3 1

1
3 2
= x − 2x − 11x + 12 dx
−3
4
2
3
− x − 2x − 11x + 12 dx
1
4 3 2 1
x 2x 11x
= − − + 12x
4 3 2 −3
4 3 2 4
x 2x 11x
− − − + 12x . (∗)
4 3 2
1
Here we are using the standard shorthand
b
F(x)| a
to stand for
F(b) − F(a).
Thus we have
160 297
(∗) = + .
3 12
Notice that, by design, each component of the area has made a positive
contribution to the final answer. The total area is then
937
Area = .
12
EXAMPLE 4.9
Calculate the (positive) area between f(x) = sin x and the x-axisfor
−2π ≤ x ≤ 2π.

SOLUTION
We observe that f(x) = sin x ≥ 0 for −2π ≤ x ≤−π and 0 ≤ x ≤ π.
Likewise, f(x) = sin x ≤ 0 for −π ≤ x ≤ 0 and π ≤ x ≤ 2π. As a result

−π 0
Area = sin xdx − sin xdx
−2π −π

π 2π
+ sin xdx − sin xdx.
0 π