4.4 The Fundamental Theorem of Calculus

1. 4.4 The Fundamental Theorem of Calculus If a function is continuous on the closed interval [a, b], then where F is any function that F’(x) = f(x) x in [a, b].

2. Examples 1 – 0 = 1

3. Integration with Absolute Value We need to rewrite the integral into two parts.

4. Ex. Find the area of the region bounded by y = 2x2 – 3x + 2, the x-axis, x = 0, and x = 2.

5. The Mean Value Theorem for Integrals If f is continuous on [a, b], then a number c in the open interval (a, b) rectangle area is equal to actual area under curve. a b a b a b Mean Value rect. Circumscribed Rect inscribed rectangle

6. Find the value c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = x3 over [0, 2]. 4 = 2c3 8 c3 = 2 2

7. Average Value If f is continuous on [a, b], then the average value of f on this interval is given by Find the average value of f(x) = 3x2 – 2x on [1, 4]. 40 16 Ave. = 16 (1,1)

8. The Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then for every x in the interval Ex. Apply the Second Fund. Thm. of Calculus

9. Find the derivative of F(x) = Apply the second Fundamental Theorem of Calculus with the Chain Rule.