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Moving from the Indefinite to the Definite Integral

Lecture 18. Moving from the Indefinite to the Definite Integral. The Definite Integral as a -- Sum : A Numerical Approach Area : A Geometric Approach. Cell Phone Cost.

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Moving from the Indefinite to the Definite Integral

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  1. Lecture 18 Moving from the Indefinite to the DefiniteIntegral The Definite Integral as a --Sum: A Numerical ApproachArea: A Geometric Approach

  2. Cell Phone Cost Your cell phone company offers you an innovative pricing scheme. When you make a call, the marginal cost of the tth call is: How much for a 60 minute phone call?

  3. Cell Phone Cost

  4. Cell Phone Cost 0.20 ($/min) * 60 (min) = $12.00

  5. Cell Phone Cost 0.20 ($/min) * 10 (min) = $2.00 0.1818 ($/min) * 10 (min) = $1.818 0.1667 ($/min) * 10 (min) = $1.667 Total cost for 60 min. = $9.79

  6. Riemann Sum If f is a continuous function, then the left Riemann sum with n equal subdivisions for f over the interval [a, b] is defined to be

  7. The Definite Integral If f is a continuous function, the definite integralof f from a to b is defined to be The function f is called the integrand, the numbers a and b are called the limits of integration, and the variable x is called the variable of integration.

  8. Approximating the Definite Integral Ex. Calculate the Riemann sum for the integral using n = 10.

  9. The Definite Integral As a Total Ex. If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by

  10. Area Under a Graph a b f continuous, nonnegative on [a, b]. The area is

  11. Geometric Interpretation (All Functions) R1 R3 a b R2 Area of R1 – Area of R2 + Area of R3

  12. Area Using Geometry Ex. Use geometry to compute the integral Area = 8 Area = 2

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