Using the fundamental theorem to find the definite
integrals
December 8, 2013
Using the fundamental theorem to find the definite integrals
Fundamental theorem of calculus
Theorem
If F 0 (t) is continuous for a ≤ t ≤ b, then
Z
b
F 0 (t)dt = F (b) − F (a)
a
In other words: The definite integral of the derivative of a function
gives the total change in the function.
Using the fundamental theorem to find the definite integrals
Example
Z
3
2xdx =?
1
Using the fundamental theorem to find the definite integrals
Example
Z
3
2xdx = 8.000
1
Using the fundamental theorem to find the definite integrals
Example
Z
b
F 0 (x)dx = F (b) − F (a)
a
Let F 0 (x) = 2x.
Using the fundamental theorem to find the definite integrals
Example
Z
b
F 0 (x)dx = F (b) − F (a)
a
Let F 0 (x) = 2x.
F (x) is an antiderivative of 2x, take F (x) = x 2 .
Using the fundamental theorem to find the definite integrals
Example
Z
b
F 0 (x)dx = F (b) − F (a)
a
Let F 0 (x) = 2x.
F (x) is an antiderivative of 2x, take F (x) = x 2 .
R3
R3 0
2
2
1 2xdx = 1 F (x)dx = F (3) − F (1) = 3 − 1 = 9 − 1 = 8
Using the fundamental theorem to find the definite integrals
Notation
b
F (x) = F (b) − F (a)
a
Using the fundamental theorem to find the definite integrals
Notation
b
F (x) = F (b) − F (a)
a
3
R3
2 .
2xdx
=
x
1
1
Using the fundamental theorem to find the definite integrals
Example
R3
0
6x 3 dx =?
Using the fundamental theorem to find the definite integrals
Example
R3
3
0 6x dx =?
R5
1 (4x + 3)dx
=?
Using the fundamental theorem to find the definite integrals
Example
R3
3
0 6x dx =?
R5
1 (4x + 3)dx
R9 √
4 5 xdx =?
=?
Using the fundamental theorem to find the definite integrals
Example
Find the area under the graph of f (x) = 4e 0.3x between x = 0 and
x =4
Using the fundamental theorem to find the definite integrals
Example
Compute
R3
0
2xe x
2 +1
dx.
Using the fundamental theorem to find the definite integrals
Example
Z
1
4
x2
dx =?
1 + x3
Using the fundamental theorem to find the definite integrals
Method
1
Compute the indefinite integral, expressing an antiderivative in
terms of the original variable, and then apply the fundamental
theorem.
2
Convert the original limits to new limits in terms of the new
variable, do not convert the antiderivative back to the original
variable, and then apply the fundamental theorem.
Using the fundamental theorem to find the definite integrals
Example
Z
2
4
√
y
dy =?
5−y
Using the fundamental theorem to find the definite integrals
Example
Z
5
(ln t)2 dt =?
3
Using the fundamental theorem to find the definite integrals