More on Volumes & Average Function Value

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Presentation transcript:

More on Volumes & Average Function Value Chapter 6.5 March 1, 2007

Average On the last test (2), the average of the test over 2 classes was: FYI - there were 71 who scored a 9 or 10, which means the median was a 9.

Average Function Value of f(x) on the interval [a,b] We can divide the interval [a,b] into n subintervals and average the selected function values.

Average function value If we let then number of points selected go to infinity We arrive at the Definite Integral!

Find the average function value of over the interval

Find the average function value of over the interval

Average function value If we let multiply both sides of the formula, we get: Thinking area: The area of the rectangle (b-a) by Avef has the same area as the area under the curve as seen…..

Find the average function value of over the interval Solving for x, we get: The area of the green rectangle = the area under over the interval

Let R be the region in the x-y plane bounded by Set up the integral for the volume of the solid obtained by rotating R about the line y = 3, a) Integrating with respect to x. b) Integrating with respect to y.

Integrating with Respect to x: Outside Radius ( R ): Inside Radius ( r ): Area: Volume:

Integrating with Respect to y: Length of Slice: Inside Radius ( r ): Area: Volume: