# Fundamental theorem of calculus II - PowerPoint PPT Presentation

1 / 17

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Fundamental theorem of calculus II

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

#### Presentation Transcript

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 Dx that break this picture. What are they?

0

STOP

Area under the curve

“Definite integral”

0

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

0

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

0

FToC: Differentiation “undoes” integration

Want

0

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

0

0

FToC: Integration “undoes” differentiation

0

0

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

0

Change of variables example: Trigonometric functions

Choose to identify

Find in integration table:

0

Integrals

Integrate

Area under the curve

Fundamental theorem of calculus I

Change of variables

Fundamental theorem of calculus II