Integration 4 Copyright © Cengage Learning. All rights reserved.

The Fundamental Theorem of Calculus Copyright © Cengage Learning. All rights reserved. 4.4 Lesson A Part 1 of 3 FTC only

3 4.4 The Fundamental THM of Calculus Warm-Up Find the indefinite integral. What does it represent?

4 Evaluate a definite integral using the Fundamental Theorem of Calculus. Objective

5 The Fundamental Theorem of Calculus

6 Let F(t) represent the position of a particle moving along a horizontal axis. At time t = 0 the position of the particle is x=3 units and the velocity of the particle units per minute. Find the position of the particle at 2 minutes. Example:

7 4.4 The Fundamental THM of Calculus Calculus Warm-Up 11/18/2010 Using the definite integral above, evaluate the following limit without using a summation.

8 The Fundamental Theorem of Calculus The two major branches of calculus: differential calculus and integral calculus. At this point, these two problems might seem unrelated—but there is a very close connection. The connection was discovered independently by Isaac Newton and Gottfried Leibniz and is stated in a theorem that is appropriately called the Fundamental Theorem of Calculus.

9 Informally, the theorem states that differentiation and (definite) integration are inverse operations, in the same sense that division and multiplication are inverse operations. To see how Newton and Leibniz might have anticipated this relationship, consider the approximations shown in Figure Figure 4.26 The Fundamental Theorem of Calculus

10 The Fundamental Theorem of Calculus The slope of the tangent line was defined using the quotient Δy/Δx. Similarly, the area of a region under a curve was defined using the product ΔyΔx. So, at least in the primitive approximation stage, the operations of differentiation and definite integration appear to have an inverse relationship in the same sense that division and multiplication are inverse operations. The Fundamental Theorem of Calculus states that the limit processes preserve this inverse relationship.

11 Definition of a definite integral: Sum of areas of very skinny rectangles under the curve

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13 The Fundamental Theorem of Calculus

14 A Geometrical Perspective Consider an object moving at a constant rate of 40 mi/hr. Since distance = rate. time, if we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.

15 Consider an object moving at a variable rate. The distance that the object travels is equal to the area under the velocity curve.

16 time velocity What was the net change in distance over the 4 second interval? Consider an object moving at a constant rate of 3 ft/sec. Since distance = rate. time If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. A Geometrical Perspective

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18 The integral of a rate of change is the net amount of change.

19 So, how do you find the net change? Fundamental Theorem of Calculus Find the accumulated value of the anti-derivative from a to b!

20 Fundamental Theorem of Calculus

21 You Try

22 You Try

23 You Try

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The Fundamental THM of Calc.

26

The Fundamental THM of Calculus

28 Caution  What’s wrong with the following?

29 How about this one:

30 It is not necessary for a function to be continuous to be integrable! The FTC guarantees that all continuous functions are integrable. But some functions with discontinuities are also integrable. A bounded function that has a finite number of points of discontinuity on an interval [a,b] will still be integrable on the interval if it is continuous everywhere else!

31 BC Homework 4.4  Pg odd

32 Homework 4.4 Day 1  4.4 Pg odd