Second Fundamental Theorem

Recall: First Fundamental Theorem of Calculus (FTC 1)
If f is continuous and F? = f, then
? b
f(x)dx = F(b) ? F(a)
a
We can also write that as ? b
x=b
f(x)dx = f(x)dx
x=a
a
Do all continuous functions have antiderivatives? Yes. However...
What about a function like this?
2
e?x dx =?? Yes, this antiderivative exists. No, it’s not a function we’ve met before: it’s a new function. The new function is defined as an integral:
x
2
F(x)= e?tdt
0
2
It will have the property that F?(x)= e?x .
sin x
1/2
Other new functions include antiderivatives of e?x2 ,x e?x2 ,, sin(x2), cos(x2),...
x
Second Fundamental Theorem of Calculus (FTC 2)
x
If F(x)= f(t)dt and f is continuous, then
a
F?(x)= f(x)
Geometric Proof of FTC 2: Use the area interpretation: F(x) equals the area under the curve between a and x.
?F = F(x +?x) ? F(x)
?F ? (base)(height) ? (?x)f(x) (See Figure 1.)
?F
?x ? f(x)
?F
Hence lim = f(x)
?x0 ?x
?
But, by the definition of the derivative:
?F
lim = F?(x)
?x0 ?x
?
1
?? ? ?
?
Lecture 20 18.01 Fall 2006
a x x+?x
F(x)
?F
y
Figure 1: Geometric Proof of FTC 2.
Therefore,
F?(x) = f(x)
Another way to prove FTC 2 is as follows:
?F 1 x+?x x
?x
=
?x
f(t)dt ? f(t)dt
a a
1
? x+?x
=
?x
f(t)dt (which is the “average value” of f on the interval x ? t ? x + ?x.)
x
As the length ?x of the interval tends to 0, this average tends to f(x).
Proof of FTC 1 (using FTC 2)
x
Start with F? = f (we assume that f is continuous). Next, define G(x) = f(t)dt. By FTC2,
a
G?(x) = f(x). Therefore, (F ? G)? = F? ? G? = f ? f = 0. Thus, F ? G = constant. (Recall we
used the Mean Value Theorem to show this).
Hence, F(x) = G(x) + c. Finally since G(a) = 0,
? b
f(t)dt = G(b) = G(b) ? G(a) = [F(b) ? c] ? [F(a) ? c] = F(b) ? F(a)
a
which is FTC 1.
Remark. In the preceding proof G was a definite integral and F could be any antiderivative. Let
us illustrate with the example f(x) = sin x. Taking a = 0 in the proof of FTC 1,
? x ?x
G(x) = cos t dt = sin t?? = sin x and G(0) = 0.
0 0
2
?
?
Lecture 20 18.01 Fall 2006
If, for example, F(x) = sin x + 21. Then F?(x) = cos x and
? b
sin x dx = F(b) ? F(a) = (sin b + 21) ? (sin a + 21) =