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5.2. THE SECOND FUNDAMENTAL THEOREM OF CALCULUS
Preview Activity 5.2. Consider the function A defined by the rule
A(x) =
x
1
f (t) dt,
where f (t) = 4 − 2t.
(a) Compute A(1) and A(2) exactly.
(b) Use the First Fundamental Theorem of Calculus to find an equivalent formula
for A(x) that does not involve integrals. That is, use the first FTC to evaluate
x
1
(4 − 2t) dt.
(c) Observe that f is a linear function; what kind of function is A?
(d) Using the formula you found in (b) that does not involve integrals, compute A ′ (x).
(e) While we have defined f by the rule f (t) = 4 − 2t, it is equivalent to say that f
is given by the rule f (x) = 4 − 2x. What do you observe about the relationship
between A and f ?
⊲⊳
The Second Fundamental Theorem of Calculus
The result of Preview Activity 5.2 is not particular to the function f (t) = 4 − 2t, nor to the
choice of “1” as the lower bound in the integral that defines the function A. For instance,
if we let f (t) = cos(t) − t and set A(x) =
x
2
f (t) dt, then we can determine a formula for
A without integrals by the First FTC. Specifically,
A(x) =
x
2
(cos(t) − t) dt
= sin(t) −
1
2
t
2
x
2
= sin(x) −
1
2
x
2 − (sin(2) − 2) .
Differentiating A(x), since (sin(2) − 2) is constant, it follows that
A
′ (x) = cos(x) − x,
and thus we see that A ′ (x) = f (x). This tells us that for this particular choice of f , A
is an antiderivative of f . More specifically, since A(2) =
2
2
f (t) dt = 0, A is the only
antiderivative of f for which A(2) = 0.