1. 1 Fundamental Theorem of Calculus Section 4.4

2. 2 The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b] then Fundamental Theorem of Calculus We express this as

3. 3 Examples Example: Calculator check>> CALC 7:

4. 4 Examples Example:

5. 5 Examples Example:

6. 6 Examples Example:

7. 7 Examples Example:

8. 8 Examples Find the area bounded by the graphs of y = x + sin x, the x-axis, x = 0, and x =

9. 9 Mean Value Theorem for Integrals If f is continuous on [a, b], then there exists a number c in the closed interval [a, b] such that Somewhere between the inscribed and circumscribed rectangles there is a rectangle whose area is precisely equal to the area under the curve. Inscribed Rectangle Mean Value Rectangle Circumscribed Rectangle

10. 10 Average Value of a Function The value of f(c) in the Mean Value Theorem for Integrals is called the average value of f on the interval [a, b]. Since then solving for f(c) gives

11. 11 Example Find the average value of f(x) = sin x on [0,  ] and all values of x for which the function equals its average value.

12. 12 The Second Fundamental Theorem of Calculus a f x Using x as the upper limit of integration.

13. 13 Example Integrate to find F as a function of x and demonstrate the Second Fundamental Theorem of Calculus by differentiating the result.

14. 14 Example Integrate to find F as a function of x and demonstrate the Second Fundamental Theorem of Calculus by differentiating the result.

15. 15 Homework Sect 4.4 page 291 #5 – 23 odd, 27, 29, 33 – 39 odd, 47