1. The Fundamental Theorem of Calculus Section 5.4

2. Newton and Leibniz make HEADLINES! Derivatives and Integrals are INVERSE operations They both discovered independently that... Just like multiplication and division are inverse operations...One UNDOES the other! If they had newspapers in their day!

3. 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 When you read/take notes on this section you will find the proof of this theorem on p. 319 in your text!

4. So... Let’s revisit some of your problems from 5.3 Instead of using geometry, let’s try following the lead of Newton and Leibniz and use the Fundamental Theorem of Calculus BUT…

5. Let’s try this out… 1.Sketch the region 2.Find the area indicated by the integral. From geometry... Area = (base)(height) = (2)(4) = 8 u 2 Look.... No C’s = 4(3) – 4(1) = 12 – 4 = 8 u 2

6. Give this one a try… 1.Sketch the region 2.Find the area indicated by the integral. From Geometry Area of a trapezoid =.5(width)(base1+ base2) = (.5)(3)(2+5) = 10.5 u 2 Widh =3 base1=2 base2 =5 =10.5 u 2

7. Try this one… Evaluate this definite integral. =2.5 u 2 Area=(.5)1(.5)=.25 + Area =(.5)1.5(3)=2.25 Area = 2.5 Think:

8. The Mean Value Theorem for Integrals If f is continuous on [a,b] then there exists a number c in [a,b] such that If f is continuous on [a,b] then there exists a number c in [a,b] such that This means that there exists a rectangle with a width of b-a and a height f(c) that can be created that has exactly the same area as the area under the curve f(x) from a to b. This means that there exists a rectangle with a width of b-a and a height f(c) that can be created that has exactly the same area as the area under the curve f(x) from a to b.

9. The Mean Value Theorem for Integrals Find the value(s) of c guaranteed by the Mean Value Theorem for Integrals for the functions over the indicated interval. Find the area of the given region with an integral: Use that area to calculate the value of c. So how do we find c?

10. HOMEWORK – more practice Wednesday, January 23 Read/take notes on 5.4 (p. 318-321) and do p. 327 #12, 15, 18, 21,..., 54 (multiples of 3) 5.4 day 1