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# Fundamental theorem of calculus II - PowerPoint PPT Presentation

Integrals. Integrate. Area under the curve. Fundamental theorem of calculus I. Change of variables. Fundamental theorem of calculus II. Area under the curve. 0. Area under the curve. Verify that this sum makes sense. There are values of D x that break this picture. What are they?. 0.

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## PowerPoint Slideshow about 'Fundamental theorem of calculus II' - pepin

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Presentation Transcript

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

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

0

STOP

“Definite integral”

0

STOP

We wrote a differential. What is coordinately shrinking with ?

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

0

STOP

Differentiation “undoes” integration. Do you remember why?

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

FToC: Differentiation “undoes” integration

Want

0

FToC: Differentiation “undoes” integration

Want

0

FToC: Differentiation “undoes” integration

Want

0

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

FToC: Integration “undoes” differentiation

0

0

FToC: Integration “undoes” differentiation

0

0

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

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II

Choose to identify

Find in integration table:

0

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II