The Area Question and the Integral

1
The Area Question and the Integral

• Lesson 6.1

2
Area Under the Curve

• What does the following demo suggest about how to
  measure the area under the curve?

3
Area Under the Curve

• Using more and more rectangles to approximate the
  area

4
The Area Under a Curve

• Divide the area underthe curve on the interval
  a,b inton equal segments
• Each "rectangle" has height f(xi)
• Each width is ?x
• The area if the i th rectangle is f(xi)?x
• We sum the areas

5
Summation Notation

• We use summation notation
• Note the basic rules and formulas
• Summation Formulas, pg 218

6
Use of Calculator

• Note again summation capability of calculator
• Syntax is ? (expression, variable, low,
  high)

7
Practice Summation

• Try these

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Limit of a Sum

• For a function f(x), the area under the
  curvefrom a to b iswhere ?x (b a)/n and
• Consider the region bounded by f(x) x2 the
  axes, and the lines x 2 and x 3

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Limit of a Sum

• Now
• So

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Limit of a Sum

• Continuing

11
Practice Summation

• For our general formula
• let f(x) 3 2x on 0,1

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The SumCalculated

• Consider the function2x2 7x 5
• Use ?x 0.1
• Let the left edgeof each subinterval
• Note the sum

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The Area Under a Curve

• The accuracy of the summation will increase if we
  have more segments
• As we increase n
• As n gets infinitely large the summation is exact

14
The Definite Integral

• We will use another notation to represent the
  limit of the summation

15
Example

• Try
• Use summation on calculator.

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Example

• Note increased accuracy with smaller ?x

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Limit of the Sum

• The definite integral is the
  limit of the sum.

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Practice

• Try this
• What is the summation?
• Which gives us
• Now take limit

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Practice

• Try this one
• What is ?x?
• What is the summation?
• For n 50?
• Now take limit

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Assignment

• Lesson 6.1
• Page 221
• Exercises 1 17 odd