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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. 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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  1. Integrals Integrate Area under the curve Fundamental theorem of calculus I Change of variables Fundamental theorem of calculus II

  2. Area under the curve 0

  3. Area under the curve Verify that this sum makes sense. There are values of Dx that break this picture. What are they? 0 STOP

  4. Area under the curve “Definite integral” 0 STOP We wrote a differential. What is coordinately shrinking with ?

  5. 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?

  6. Integrals Integrate Area under the curve Fundamental theorem of calculus I Change of variables Fundamental theorem of calculus II

  7. FToC: Differentiation “undoes” integration Want 0

  8. FToC: Differentiation “undoes” integration Want 0

  9. FToC: Differentiation “undoes” integration Want 0

  10. Integrals Integrate Area under the curve Fundamental theorem of calculus I Change of variables Fundamental theorem of calculus II

  11. FToC: Integration “undoes” differentiation 0 0

  12. FToC: Integration “undoes” differentiation 0 0

  13. 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

  14. Integrals Integrate Area under the curve Fundamental theorem of calculus I Change of variables Fundamental theorem of calculus II

  15. Change of variables 0

  16. Change of variables example: Trigonometric functions Choose to identify Find in integration table: 0

  17. Integrals Integrate Area under the curve Fundamental theorem of calculus I Change of variables Fundamental theorem of calculus II

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