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Fundamental theorem of calculus IIPowerPoint Presentation

Fundamental theorem of calculus II

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Fundamental theorem of calculus II

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Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

Area under the curve

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Area under the curve

Verify that this sum makes sense. There are values of Dx that break this picture. What are they?

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STOP

Area under the curve

“Definite integral”

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STOP

We wrote a differential. What is coordinately shrinking with ?

Example: Area under a line

If we hold a in place, the derivative of A “happens” to be

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STOP

Differentiation “undoes” integration. Do you remember why?

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

FToC: Differentiation “undoes” integration

Want

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FToC: Differentiation “undoes” integration

Want

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FToC: Differentiation “undoes” integration

Want

0

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

FToC: Integration “undoes” differentiation

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FToC: Integration “undoes” differentiation

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Example integral table

Generic differentiation rule

Notion of anti-derivative:Instead of maligning the indefinite integral as the result of “forgetting” to write down symbols in a definite integral, one often says that, in the context of an equation lacking beginning and end points, such as , the “curvy S” indicates merely that taking the derivative of gives . This kind of use of language does not require discussion of the notion of area under a curve.

STOP

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

Change of variables

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Change of variables example: Trigonometric functions

Choose to identify

Find in integration table:

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Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II