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?

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

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

Generalizing In general … The actual area is where a b In general … The actual area is where Try Java Applet 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)

Practice Summation Try these

Limit of a Sum a b For a function f(x), the area under the curve from a to b is where x = (b – a)/n and Consider the region bounded by f(x) = x2 the axes, and the lines x = 2 and x = 3

Limit of a Sum Now So

Limit of a Sum Continuing …

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

Assignment Lesson 5.2 Page 303 Exercises 1 – 61 EOO (omit 45)