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warm up

warm up. 1.Find the derivative using the quotient rule. 2. Find the derivative using the chain rule. Lesson 15-3 The Area Under the Curve . Objective: To find areas under graphs of polynomial functions To find values of integrals of polynomial functions.

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warm up

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  1. warm up • 1.Find the derivative using the quotient rule. • 2. Find the derivative using the chain rule.

  2. Lesson 15-3 The Area Under the Curve Objective: To find areas under graphs of polynomial functions To find values of integrals of polynomial functions

  3. How do we find the area between a curve and the x-axis from x = a to x = b? • We can approximate it with rectangles

  4. The Area Problem

  5. The Area Problem • The approximation is improved by increasing the number of rectangles. • The width of each rectangle is:

  6. Area Under the Curve • The area of the 1st rectangle is • The Total Area = • in order for the width to approach 0, n approaches infinity • This is also the definite integral:

  7. Definite Integral where This process is called integration.

  8. Area Under the Curve • Example: Find the area under the curve from x=0 to x =1 when y = x2.

  9. Area Under the Curve

  10. Area Under the Curve

  11. Sums of Series • 1+2+3+…n= • 12+22+32+…n2= • 13+23+33+…n3=

  12. Example • Use limits to fin the area of the region between the graph of and the x axis from x = 2 to x = 4. That is find • To do this use

  13. Sources • "The area under a curve." Preview of Calculus. LeTourneau University, n.d. Web. 3 Apr. 2014. <http://www.letu.edu/.../Lesson2.1A%20Preview%20of...‎>.

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