Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 1)

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Presentation transcript:

Chapter 6 Integration Section 5 The Fundamental Theorem of Calculus (Day 1)

2 Barnett/Ziegler/Byleen Business Calculus 12e Learning Objectives for Section 6.4 The Definite Integral The student will be able to: Find the area under a curve using a definite integral.

3 Review  We learned in the first part of this lesson that we can estimate the area under a curve over an interval [a, b] by breaking the interval into rectangles and summing up the areas of these rectangles. Barnett/Ziegler/Byleen Business Calculus 12e f (x) = 0.5 x Error = 12Error = 3 If you increase the number of subintervals (rectangles) the area approaches the actual area under the curve.

4 Barnett/Ziegler/Byleen Business Calculus 12e Limit of Riemann Sums Theorem 3. Let f be a continuous function on [a, b], then the Riemann sums for f on [a, b] approach a real number limit I as n  . If the number of subintervals approaches infinity, the sum will be the actual area under the curve.

5 Barnett/Ziegler/Byleen Business Calculus 12e The Definite Integral This limit I of the Riemann sums for f on [a, b] is called the definite integral of f from a to b, denoted The integrand is f (x), the lower limit of integration is a, and the upper limit of integration is b. This definite integral can be used to determine the actual area between the graph of f(x) and the x-axis!

6 Barnett/Ziegler/Byleen Business Calculus 12e Negative Values If f (x) is positive for some values of x on [a, b] and negative for others, then the definite integral symbol represents the cumulative sum of the signed areas between the graph of f (x) and the x axis, where areas above are positive and areas below negative. y = f (x) a bA B

7 Definite Integrals Barnett/Ziegler/Byleen Business Calculus 12e f (x) = 0.5 x The approximate area: The actual area is represented by: f (x) = 0.5 x

8 Evaluating Definite Integrals  Evaluating a definite integral is like evaluating an indefinite integral except for one step.  We will use what is called the Fundamental Theorem of Calculus! Barnett/Ziegler/Byleen Business Calculus 12e

9 Fundamental Theorem of Calculus If f is a continuous function on the closed interval [a, b], and F is any antiderivative of f, then

10 Barnett/Ziegler/Byleen Business Calculus 12e Example 1 Evaluate: Plug in the interval values and subtract. Integrate like before. Notation is different & no more +C

11 Barnett/Ziegler/Byleen Business Calculus 12e Example 2 Evaluate:

12 Example 3  Find the exact area under the curve from [1,5]. Barnett/Ziegler/Byleen Business Calculus 12e f (x) = 0.5 x

13 Example 3 (continued) Barnett/Ziegler/Byleen Business Calculus 12e

14 Barnett/Ziegler/Byleen Business Calculus 12e Example 4 Let u = 2x Evaluate: du = 2 dx

15 Barnett/Ziegler/Byleen Business Calculus 12e Example 5 Evaluate:

16 Barnett/Ziegler/Byleen Business Calculus 12e Example 6 Evaluate:

17 Barnett/Ziegler/Byleen Business Calculus 12e Example 7 Evaluate:

18 Barnett/Ziegler/Byleen Business Calculus 12e Definite Integral Properties

19 Homework Barnett/Ziegler/Byleen Business Calculus 12e #6-5A Pg. 401 (6-39 mult of 3, 49, 51)