Fundamental theorem of calculus II

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