Calculus 5.4

1
Fundamental Theorem of Calculus
Morro Rock, California
2
The Fundamental Theorem of Calculus, Part 1
If f is continuous on , then the
function
has a derivative at every point in , and
3
First Fundamental Theorem
4
First Fundamental Theorem
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
5
First Fundamental Theorem
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
6
First Fundamental Theorem
New variable.
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
7
The long way
First Fundamental Theorem
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
8
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
9
(No Transcript)
10
We can change the sign of the integral and
reverse the limits.
11
We split the integral into two parts.
It does not matter what constant we use!
(Limits are reversed.)
(Chain rule is used.)
12
The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of ,
and if F is any antiderivative of f on
, then
(Also called the Integral Evaluation Theorem)
We already know this!
To evaluate an integral, take the
anti-derivatives and subtract.
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