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More on Volumes & Average Function Value

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Average On the last test (2), the average of the test was: FYI - there were 35 who scored a 9 or 10, which means the median was a 9.

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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.

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

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Find the average function value of over the interval

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

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

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Let R be the region in the x-y plane bounded by Set up the integral to find the area of the region. Top Function: Bottom Function: Area: Bounds: [0,2] Length:

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Set up the integral to find the volume of the solid whose base is the region between the curves: and with cross sections perpendicular to the x-axis equilateral triangles. Area : Volume: Length:

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Find the volume of the solid generated by revolving the region defined by about the line y = 3. Area : Volume: Length: Inside (r) : Outside (R) :

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Solids of Revolution If there is a gap between the function and the axis of rotation, we have a washer and use: If there is NO gap, we have a disk and use: Volume =

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Note in the disk/washer methods, the focus in on the radius (perpendicular to the axis of rotation) and the shape it forms. We can also look at a slice that is parallel to the axis of rotation.

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Area: Length of slice

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Volume = Slice is PARALLEL to the AOR

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Using on the interval [0,2] revolving around the x-axis using planar slices PARALLEL to the AOR, we find the volume: Radius? Length of slice? Area? Volume?

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Back to example: Find volume of the solid generated by revolving the region about the y-axis using cylindrical slices Length of slice ( h ): Radius ( r ): Area: Volume:

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Find the volume of the solid generated by revolving the region: about the y-axis, using cylindrical slices. Length of slice ( h ): Inside Radius ( r ): Area: Volume:

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Find the volume of the solid generated by revolving the region defined by about the line x = 3. Area : Volume: Length: Radius :

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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.

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

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

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6.3 Volumes by Cylindrical Shells. Find the volume of the solid obtained by rotating the region bounded,, and about the y -axis. We can use the washer.

6.3 Volumes by Cylindrical Shells. Find the volume of the solid obtained by rotating the region bounded,, and about the y -axis. We can use the washer.

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