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4.3 Fundamental Theorem of Calculus

# 4.3 Fundamental Theorem of Calculus - fnInt abs f x x a b...

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The Fundamental Theorem of Calculus Section 4.3

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The Fundamental Thm. Of Calculus, Part 1 ( ) ( ) = x a g x f t dt a x b
The Fundamental Thm. Of Calculus, Part 2 If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], then ( ) ( ) ( ). b a f x dx F b F a = -

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The following notation is convenient to use when evaluating a definite integral: It is not necessary to include a constant of integration C because it will always “cancel out” in a definite integral. a b f(x) dx = F(x) a b
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.

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How to Find Total Area Numerically To find the area between the graph of y = f(x) and the x-axis over the interval [ a, b ] numerically, evaluate ( ( ( )), , , ).

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Unformatted text preview: fnInt abs f x x a b When solving a definite integral involving absolute value, rewrite the integral in 2 parts, splitting it up at the critical pt. A slightly different situation is when the variable x is used as the upper limit of integration. To avoid the confusion of using x in 2 different ways, we temporarily switch to using t as the varable of integration. F(x) = a x f (t) dt constant F is a function of x. f is a function of t. When upper limit of integration is a variable other than plain x , combine the Fund. Thm., Part 1 with the Chain Rule when finding the derivative of an integral. It really amounts to just plugging in the upper limit and multiplying by its derivative. Joke Time Why was the elephant standing on the marshmallow? He didn’t want to fall in the hot chocolate! How do you stop an elephant from charging? Take away his credit card....
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