James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 5.3 The Fundamental Theorem of Calculus 397
Although the theorem may be surprising at first glance, it becomes plausible if we
interpret it in physical terms. If vstd is the velocity of an object and sstd is its position at
time t, then vstd − s9std, so s is an antiderivative of v. In Section 5.1 we considered an
object that always moves in the positive direction and made the guess that the area under
the velocity curve is equal to the distance traveled. In symbols:
y b
vstd dt − ssbd 2 ssad
a
That is exactly what FTC2 says in this context.
Example 5 Evaluate the integral y 3
e x dx.
1
Compare the calculation in Example 5
with the much harder one in Example
5.2.3.
SOLUTION The function f sxd − e x is continuous everywhere and we know that an antiderivative
is Fsxd − e x , so Part 2 of the Fundamental Theorem gives
y 3
e x dx − Fs3d 2 Fs1d − e 3 2 e
1
Notice that FTC2 says we can use any antiderivative F of f. So we may as well use
the simplest one, namely Fsxd − e x , instead of e x 1 7 or e x 1 C.
n
We often use the notation
So the equation of FTC2 can be written as
Fsxdg a
b
− Fsbd 2 Fsad
y b
b
f sxd dx − Fsxdg
a
a
where
F9− f
b
Other common notations are Fsxd
|a
and fFsxdg ab .
Example 6 Find the area under the parabola y − x 2 from 0 to 1.
SOLUTION An antiderivative of f sxd − x 2 is Fsxd − 1 3 x 3 . The required area A is found
using Part 2 of the Fundamental Theorem:
In applying the Fundamental Theorem
we use a particular antiderivative F
of f . It is not necessary to use the
most general antiderivative.
A − y 1
x 2 dx − x 3
0
3G0
1
− 13
3 2 03
3 − 1 3
If you compare the calculation in Example 6 with the one in Example 5.1.2, you will
see that the Fundamental Theorem gives a much shorter method.
n
Example 7 Evaluate y 6 dx
3 x .
SOLUTION The given integral is an abbreviation for
y 6
3
1
x dx
An antiderivative of f sxd − 1yx is Fsxd − ln | x |
Fsxd − ln x. So
y 6
3
1
x dx − ln xg 6 3 − ln 6 2 ln 3 − ln 6 3 − ln 2
and, because 3 < x < 6, we can write
n
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