Page 122 - Calculus Demystified

CHAPTER 4
SOLUTION The Integral 109
2
We use the Fundamental Theorem. In this example, f(x) = x . We need to
find an antiderivative F. From our experience in Section 4.1, we can determine
3
that F(x) = x /3 will do. Then, by the Fundamental Theorem of Calculus,
2 3 3

2 2 0 8
x dx = F(2) − F(0) = − = .
0 3 3 3
Notice that this is the same answer that we obtained using Riemann sums in
Example 4.4.
EXAMPLE 4.6
Calculate

π
sin xdx.
0
SOLUTION
In this example, f(x) = sin x. An antiderivative for f is F(x) =− cos x.
Then

π
sin xdx = F(π) − F(0) = (− cos π) − (− cos 0) = 1 + 1 = 2.
0
EXAMPLE 4.7
Calculate

2
x
3
e − cos2x + x − 4xdx.
1
SOLUTION
x
3
In this example, f(x) = e − cos 2x + x − 4x. An antiderivative for f is
4
2
x
F(x) = e − (1/2) sin 2x + x /4 − 2x . Therefore
2

3
x
e − cos 2x + x − 4xdx = F(2) − F(1)
1
4
2 1 2 2
= e − sin(2 · 2) + − 2 · 2
2 4
4
1 1
1 2
− e − sin(2 · 1) + − 2 · 1
2 4
1 9
2
= (e − e) − [sin 4 − sin 2]− .
2 4