Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 6.

Question

y=√1-x

x=0

y=0

Answer 1

The usual volume generated by a curve rotated around a y axis is the integral of π(r_yo²- r_yi²) dx where the r_y are the inner and outer radii of one of the annular regions that are being summed. It's a good idea to plot the function - for example, rotation around the x-axis would use the disk method.

The integral will be from x =0 to x=1

The outer radius is r = 6

The inner radius is r = 6 - sqrt(1-x)

Volume is the integral from 0 to 1 of π(36 - (6-sqrt(1-x))²)dx

or integral from 0 to 1 of π(12sqrt(1-x) - (1-x))dx

π[-8(1-x)^3/2 - x + x²/2 ] from 0 to 1

π(-1/2 + 8) = (15/2)π

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