1. 4.4
The Fundamental
Theorem of Calculus
If a function is continuous on the closed interval [a, b],
then
where F is any function that F’(x) = f(x) x in [a, b].

2. Examples
1 – 0 = 1

3. Integration with Absolute Value
We need to rewrite the integral
into two parts.

4. Ex. Find the area of the region bounded by y = 2x2 – 3x + 2,
the x-axis, x = 0, and x = 2.

5. The Mean Value Theorem for Integrals
If f is continuous on [a, b], then a number c in the
open interval (a, b)
rectangle area is equal
to actual area under curve.
a b
a b
a b
inscribed rectangle
Mean Value rect.
Circumscribed Rect

6. Find the value c guaranteed by the Mean Value Theorem for
Integrals for the function f(x) = x3 over [0, 2].
4 = 2c3 8
c3 = 2 2

7. Average Value
If f is continuous on [a, b], then the average value of
f on this interval is given by
Find the average value of f(x) = 3x2 – 2x on [1, 4]. 40 16
Ave. = 16
(1,1)

8. The Second Fundamental Theorem
of Calculus
If f is continuous on an open interval I containing a, then
for every x in the interval
Ex. Apply the Second Fund. Thm. of Calculus

9. Find the derivative of F(x) =
Apply the second Fundamental Theorem of Calculus
with the Chain Rule.