Area as the Limit of a Sum

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# Area as the Limit of a Sum - PowerPoint PPT Presentation

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

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]

We estimate with 4 rectangles using the right endpoints

x

1 2 3 4 5

Area under the Curve

We can improve our estimate by increasing the number of rectangles

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

a b

Generalizing
• In general …
• The actual area is
• where

Try Geogebra Demo

Summation Notation
• We use summation notation
• Note the basic rules and formulas
• Examples pg. 295
• Theorem 5.2 Formulas, pg 296
Use of Calculator
• Note again summation capability of calculator
• Syntax is: (expression, variable, low, high)
Finding Area by Limit Definition
• Consider the area under the curve x3 from x = 0 to x = 1
• Area

Right endpoints

Practice Summation
• For our general formula:
• let f(x) = 3 – 2x on [0,1]
Assignment
• Lesson 5.2
• Page 303
• Exercises 1 – 61 EOO