AP Calculus AB
5.4 The Fundamental Theorem of Calculus, Part 2
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Name__________________________________
I can apply the Fundamental Theorem of Calculus
I understand the relationship between the derivative and definite integral as expressed in both parts
of the Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of  a, b , and F is any antiderivative of f on  a, b , then
 f  x  dx  F b   F  a 
b
a
This part of the FTC is also called the Integration Evaluation Theorem
Proof:
Let F  x  be any antiderivative of f.
 f t  dt  F  x   C
(2) We know that  f  t  dt  F  a   C
(3) But  f  t  dt  0
x
(1) Then
a
a
a
a
a
(4) So 0  F  a   C  C   F  a 
(5) Substituting the conclusion from (4) into (1)   f  t  dt  F  x   F  a 
x
a
(6) Since x is an arbitrary value  a, b , we can conclude
 f t  dt  F b   F  a 
b
a
What this theorem tells us is that the definite integral of any continuous function f can be calculated without
taking limits, calculating Riemann Sums, or any of the other tedious methods – as long as an antiderivative
of f can be found.
Example 1 – Evaluating Integrals
Evaluate the following integrals using antiderivatives

4
2
x dx
 x
3
1
3
 1 dx


0
sin x dx
* Note that in the above examples we are evaluating integrals using antiderivatives, which is different than
finding the antiderivative . . .
Area
We already know that the definite integral can be interpreted as the net (signed) area between the graph of a
function and the x-axis. We can therefore compute areas using antiderivatives, but we must again be careful
to distinguish net (signed) area from total area. In our textbook, the word “area” will be taken to mean total
area.
How to Find Total Area Analytically
To find the area between the graph of y  f  x  and the x-axis over the interval  a, b analytically:
1. Partition  a, b with the zeros of f
2. Integrate f over each subinterval
3. Add the absolute values of the integrals
How to Find Total Area Numerically (with your calculator)
To find the area between the graph of y  f  x  and the x-axis over the interval  a, b numerically
1. Evaluate
 f  x  dx
b
a
Example 2
Find the area of the region between the curve y  4  x 2 , 0  x  3 , and the x-axis analytically
NO CALCULATOR!
Example 3
Find the area of the region between the curve y  x cos 2 x , 3  x  3 , and the x-axis using your calculator
Practice 1
Evaluate each integral using FTC part 2. Check your answer using your calculator.
1.

4
2.

3
3.

4
4.
0
1
1


0
3
x 2 dx
3
dx
x2
1
dx
x
3
4sec x tan x dx
Practice 2