254
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
Theorem of Calculus (FTC) summarizes these observations.
The Fundamental Theorem of Calculus: If f is a continuous function on [a, b],
and F is any antiderivative of f , then
b
a
f (x) dx = F(b) − F(a).
A common alternate notation for F(b) − F(a) is
F(b) − F(a) = F(x)|
b
a ,
where we read the righthand side as “the function F evaluated from a to b.” In this
notation, the FTC says that
b
a
f (x) dx = F(x)|
b
a .
The FTC opens the door to evaluating exactly a wide range of integrals. In particular,
if we are interested in a definite integral for which we can find an antiderivative F for the
integrand f , then we can evaluate the integral exactly. For instance since
d
dx [
1
3 x 3 ] = x 2 ,
the FTC tells us that
1
0
x
2 dx =
1
3
x
3
1
0
=
1
3
(1)
3 −
1
3
(0)
3
=
1
3
.
But finding an antiderivative can be far from simple; in fact, often finding a formula for
an antiderivative is very hard or even impossible. While we can differentiate just about
any function, even some relatively simple ones don’t have an elementary antiderivative. A
significant portion of integral calculus (which is the main focus of second semester college
calculus) is devoted to understanding the problem of finding antiderivatives.
Activity 4.10.
Use the Fundamental Theorem of Calculus to evaluate each of the following integrals
exactly. For each, sketch a graph of the integrand on the relevant interval and write
one sentence that explains the meaning of the value of the integral in terms of the (net
signed) area bounded by the curve.
(a)
4
−1
(2 − 2x) dx
(b)
π
2
0
sin(x) dx
(c)
1
0
e
x dx
4.4. THE FUNDAMENTAL THEOREM OF CALCULUS
Theorem of Calculus (FTC) summarizes these observations.
The Fundamental Theorem of Calculus: If f is a continuous function on [a, b],
and F is any antiderivative of f , then
b
a
f (x) dx = F(b) − F(a).
A common alternate notation for F(b) − F(a) is
F(b) − F(a) = F(x)|
b
a ,
where we read the righthand side as “the function F evaluated from a to b.” In this
notation, the FTC says that
b
a
f (x) dx = F(x)|
b
a .
The FTC opens the door to evaluating exactly a wide range of integrals. In particular,
if we are interested in a definite integral for which we can find an antiderivative F for the
integrand f , then we can evaluate the integral exactly. For instance since
d
dx [
1
3 x 3 ] = x 2 ,
the FTC tells us that
1
0
x
2 dx =
1
3
x
3
1
0
=
1
3
(1)
3 −
1
3
(0)
3
=
1
3
.
But finding an antiderivative can be far from simple; in fact, often finding a formula for
an antiderivative is very hard or even impossible. While we can differentiate just about
any function, even some relatively simple ones don’t have an elementary antiderivative. A
significant portion of integral calculus (which is the main focus of second semester college
calculus) is devoted to understanding the problem of finding antiderivatives.
Activity 4.10.
Use the Fundamental Theorem of Calculus to evaluate each of the following integrals
exactly. For each, sketch a graph of the integrand on the relevant interval and write
one sentence that explains the meaning of the value of the integral in terms of the (net
signed) area bounded by the curve.
(a)
4
−1
(2 − 2x) dx
(b)
π
2
0
sin(x) dx
(c)
1
0
e
x dx