Page 123 - Calculus Demystified

110
You Try It: Calculate the integral CHAPTER 4 The Integral
−1

3
x − cos x + xdx.
−3
Math Note: Observe in this last example, in fact in all of our examples, you can
use any antiderivative of the integrand when you apply the Fundamental Theorem
x
of Calculus. In the last example, we could have taken F(x) = e − (1/2) sin 2x +
4
2
x /4−2x +5 and the same answer would have resulted. We invite you to provide
the details of this assertion.
Justification forthe Fundamental Theorem Let f be a continuous function on
the interval [a, b]. Define the area function F by
F(x) = area under f , above the x-axis, and between 0 and x.

Fig. 4.9
Let us use a pictorial method to calculate the derivative of F. Refer to Fig. 4.9
as you read on. Now
F(x + h) − F(x) [area between x and x + h, below f ]
=
h h
f(x) · h
≈
h
= f(x).

As h → 0, the approximation in the last display becomes nearer and nearer to
equality. So we find that

F(x + h) − F(x)
lim = f(x).
h→0 h

But this just says that F (x) = f(x).
What is the practical significance of this calculation? Suppose that we wish to
calculate the area under the curve f , above the x-axis, and between x = a and
x = b. Obviously this area is F(b) − F(a). See Fig. 4.10. But we also know that