FDWK_3ed_Ch05_pp262-319

Section 5.4 Fundamental Theorem of Calculus 299
EXPLORATION 2
The Effect of Changing a in x a ftdt
The first part of the Fundamental Theorem of Calculus asserts that the derivative of
x
f t dt is f x, regardless of the value of a.
a
1. Graph NDER NINT x 2 , x, 0,x.
2. Graph NDER NINT x 2 , x, 5,x.
3. Without graphing, tell what the x-intercept of NINT x 2 , x, 0,x is. Explain.
4. Without graphing, tell what the x-intercept of NINT x 2 , x, 5,x is. Explain.
5. How does changing a affect the graph of y ddx x
f t dt
a
6. How does changing a affect the graph of y x
f t dt
a
Fundamental Theorem, Part 2
The second part of the Fundamental Theorem of Calculus shows how to evaluate definite
integrals directly from antiderivatives.
THEOREM 4 (continued)
The Fundamental Theorem of
Calculus, Part 2
If f is continuous at every point of a, b, and if F is any antiderivative of f on
a, b, then
b
f x dx Fb Fa.
a
This part of the Fundamental Theorem is also called the Integral Evaluation
Theorem.
Proof
namely
Part 1 of the Fundamental Theorem tells us that an antiderivative of f exists,
Gx x
f t dt.
Thus, if F is any antiderivative of f, then Fx Gx C for some constant C (by
Corollary 3 of the Mean Value Theorem for Derivatives, Section 4.2).
Evaluating Fb Fa, we have
Fb Fa [Gb C ] [Ga C ]
Gb Ga
b
a
b
a
b
a
a
f t dt a
f t dt
f t dt 0
f t dt.
a
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