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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].

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area under the curve
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

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

area under the curve4

x

1 2 3 4 5

Area under the Curve

We can improve our estimate by increasing the number of rectangles

area under the curve5
Area under the Curve
  • Increasing the number of rectangles to n
  • This can be done on the calculator:
generalizing

a b

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

Try Geogebra Demo

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

Right endpoints

practice summation11
Practice Summation
  • For our general formula:
  • let f(x) = 3 – 2x on [0,1]
assignment
Assignment
  • Lesson 5.2
  • Page 303
  • Exercises 1 – 61 EOO