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Fundamental theorem of calculus II PowerPoint PPT Presentation


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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. 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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Fundamental theorem of calculus II

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Fundamental theorem of calculus ii

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II


Fundamental theorem of calculus ii

Area under the curve

0


Fundamental theorem of calculus ii

Area under the curve

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

0

STOP


Fundamental theorem of calculus ii

Area under the curve

“Definite integral”

0

STOP

We wrote a differential. What is coordinately shrinking with ?


Fundamental theorem of calculus ii

Example: Area under a line

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

0

STOP

Differentiation “undoes” integration. Do you remember why?


Fundamental theorem of calculus ii

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II


Fundamental theorem of calculus ii

FToC: Differentiation “undoes” integration

Want

0


Fundamental theorem of calculus ii

FToC: Differentiation “undoes” integration

Want

0


Fundamental theorem of calculus ii

FToC: Differentiation “undoes” integration

Want

0


Fundamental theorem of calculus ii

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II


Fundamental theorem of calculus ii

FToC: Integration “undoes” differentiation

0

0


Fundamental theorem of calculus ii

FToC: Integration “undoes” differentiation

0

0


Fundamental theorem of calculus ii

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


Fundamental theorem of calculus ii

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II


Fundamental theorem of calculus ii

Change of variables

0


Fundamental theorem of calculus ii

Change of variables example: Trigonometric functions

Choose to identify

Find in integration table:

0


Fundamental theorem of calculus ii

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II


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