Second Fundamental Theorem of Calculus

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Second Fundamental Theorem of Calculus

• 5.4

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The Fundamental Theorem of Calculus, Part 2
If f is continuous on , then the
function
has a derivative at every point in , and
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Second Fundamental Theorem
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First Fundamental Theorem
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
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First Fundamental Theorem
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
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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.
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The long way
Second Fundamental Theorem
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
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1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.
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We can change the sign of the integral and
reverse the limits.
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The Fundamental Theorem of Calculus, Part 1
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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