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Area and the Definite Integral

Area and the Definite Integral. Lesson 7.3A. a. b. The Area Under a Curve. Divide the area under the curve on the interval [a,b] into n equal segments Each "rectangle" has height f(x i ) Each width is x The area if the i th rectangle is f(x i ) • x We sum the areas. •.

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Area and the Definite Integral

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  1. Area and the Definite Integral Lesson 7.3A

  2. a b 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 threctangle is f(xi)•x • We sum the areas •

  3. The SumCalculated • Consider the function2x2 – 7x + 5 • Use x = 0.1 • Let the = left edgeof each subinterval • Note the sum

  4. 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

  5. Upper limit of integration The integrand Lower limit of integration The Definite Integral • We will use another notation to represent the limit of the summation

  6. Example • Try • Use summation on calculator.

  7. Example • Note increased accuracy with smaller x

  8. Limit of the Sum • The definite integral is the limit of the sum.

  9. Practice • Try this • What is the summation? • Where • Which gives us • Now take limit

  10. Practice • Try this one • What is x? • What is the summation? • For n = 50? • Now take limit

  11. Assignment • Lesson 7.3A • Page 458 • Exercises 6 – 20 all

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