Section 15-4
The Fundamental Theorem of Calculus

If F(x) is the antiderivative of the continuous
function f(x), then
z
b

a
f ( x ) dx
= F(b) – F(a)




Evaluate
z
5
4
x
 6
dx
5
1
5
Solution: F(x) = x + C
F(b) – F(a) =
1
5
5
(5) -
1
5
5
(-6) = 10901/5




Evaluate each indefinite integral.
z
z
5
3x dx
1
2
Answer
+c
2
3
4
x
6
(3x + 5x + 3x +2x +1) dx
Answer:
3
5
5
x +
5
4
4
3
2
x +x+x +x+C


What if you were asked to find the area enclosed by two figures?
2
Example: y= x +11 (this is a line) and y=(x-4) + 3 (this is a
parabola). First graph them to get a visual.
We solve the system
of equations and see that
1
2
they intersect at (1,12) and (8,19).
(8,19)
z
8
x  11 - ((x-4) +3 dx =
1
(1,12)
z
8
= 343/6 units
2
2
1
2
9 x -x -8 dx =
9
2
2
x -
1
3
3
x -8x
 Section
15.4
 Pg 973 # 15-43 odd:
Area worksheet