6.4 Cylindrical Shells Thurs Dec 17 Revolve the region bounded by y = x^2 + 2, the x-axis, the y-axis, and x = 2 about the line x = 2.

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Presentation transcript:

6.4 Cylindrical Shells Thurs Dec 17 Revolve the region bounded by y = x^2 + 2, the x-axis, the y-axis, and x = 2 about the line x = 2

Shell Method The shell method is an alternative way to find the volume of a revolved region. This integral splits the volume into thin hollow cylinders (like the layers of a roll of tape)

Shell Method Area of a shell = 2pi(radius)(height of shell) Depending on how the region is rotated, we would use x or y Radius is usually x or y Height of shell is usually f(x) or g(y)

Ex Find the volume V of the solid obtained by rotating the region under the graph of f(x) = 1 – 2x + 3x^2 – 2x^3 over [0,1] about the y-axis

Ex2 Find the volume V obtained by rotating the region enclosed by the graphs of f(x) = x(5-x) and g(x) = 8 –x(5-x) about the y-axis

Ex3 Compute the volume V obtained by rotating the region under y = 9 – x^2 over [0,3] about the x-axis

Closure Find the volume of the region obtained by rotating the region between f(x) = x^3 and the x-axis over the interval [0,1] about the line x = 2 using the Shell Method P.389 #