5.2. THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
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In general, if f is any continuous function, and we define the function A by the rule
A(x) =
x
c
f (t) dt,
where c is an arbitrary constant, then we can show that A is an antiderivative of f . To
see why, let’s demonstrate that A ′ (x) = f (x) by using the limit definition of the derivative.
Doing so, we observe that
A
′ (x) = lim
h→0
A(x + h) − A(x)
h
= lim
h→0
x+h
c
f (t) dt −
x
c
f (t) dt
h
= lim
h→0
x+h
x
f (t) dt
h
,
(5.3)
where Equation (5.3) in the preceding chain follows from the fact that
x
c
f (t) dt +
x+h
x
f (t) dt =
x+h
c
f (t) dt. Now, observe that for small values of h,
x+h
x
f (t) dt ≈ f (x) · h,
by a simple left-hand approximation of the integral. Thus, as we take the limit in
Equation (5.3), it follows that
A
′ (x) = lim
h→0
x+h
x
f (t) dt
h
= lim
h→0
f (x) · h
h
= f (x).
Hence, A is indeed an antiderivative of f . In addition, A(c) =
c
c
f (t) dt = 0. The
preceding argument demonstrates the truth of the Second Fundamental Theorem of
Calculus, which we state as follows.
Theorem. (Second FTC) If f is a continuous function and c is any constant, then f
has a unique antiderivative A that satisfies A(c) = 0, and that antiderivative is given
by the rule A(x) =
x
c
f (t) dt.
Activity 5.4.
Suppose that f is the function given in Figure 5.10 and that f is a piecewise function
whose parts are either portions of lines or portions of circles, as pictured. In addition,
let A be the function defined by the rule A(x) =
x
2
f (t) dt.
(a) What does the Second FTC tell us about the relationship between A and f ?
(b) Compute A(1) and A(3) exactly.