How do you find the volume of the bounded region if #y = sinx#, #y = 0# from #x = pi/4#, #x = 3pi/4#, revolved around the y-axis?

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Answer 1 · 2015-06-19T00:17:26

Use the shell method.

Representative shell has volume: #2pi r h * "thickness"#

#r = x#, #h = sinx# and #"thickness" = dx#

So the volume of the solid is given by:

#V = int_(pi/4)^((3pi)/4) 2pi x sinx dx#

#V =2pi( sinx-xcosx)]_(pi/4)^((3pi)/4#

Then use trigonometry and arithmetic to get:

#V =2pi* (pi sqrt2)/2 = pi^2 sqrt2#