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Area as the Limit of a Sum

Area as the Limit of a Sum. Lesson 5.2. Area Under the Curve. What does the following demo suggest about how to measure the area under the curve?. x. 1 2 3 4 5. Area under f(x) = ln x. Consider the task to compute the area under a curve f(x) = ln x on interval [1,5].

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Area as the Limit of a Sum

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1. Area as the Limit of a Sum Lesson 5.2

2. Area Under the Curve • What does the following demo suggest about how to measure the area under the curve?

3. x 1 2 3 4 5 Area under f(x) = ln x • Consider the task to compute the area under a curve f(x) = ln x on interval [1,5] We estimate with 4 rectangles using the right endpoints

4. x 1 2 3 4 5 Area under the Curve We can improve our estimate by increasing the number of rectangles

5. Area under the Curve • Increasing the number of rectangles to n • This can be done on the calculator:

6. a b Generalizing • In general … • The actual area is • where Try Geogebra Demo

7. Summation Notation • We use summation notation • Note the basic rules and formulas • Examples pg. 295 • Theorem 5.2 Formulas, pg 296

8. Use of Calculator • Note again summation capability of calculator • Syntax is: (expression, variable, low, high)

9. Practice Summation • Try these

10. Finding Area by Limit Definition • Consider the area under the curve x3 from x = 0 to x = 1 • Area Right endpoints

11. Practice Summation • For our general formula: • let f(x) = 3 – 2x on [0,1]

12. Assignment • Lesson 5.2 • Page 303 • Exercises 1 – 61 EOO

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