Saturday, January 21, 2012

Can someone help me with cylindrical shells method to solve this question?

Find the volume of the hemisphere that results when the region in the first quadrant enclosed by the circle x^2+y^2=r^2 is revolved about the x-axis.

Can someone help me with cylindrical shells method to solve this question?
Volume

= 4 [2π ∫(x=0 to x=r) xy dy]

= 8π ∫(x=0 to x=r) x √(r2 - x2) dx

Let r2 - x2 = t2

=%26gt; -2x dx = 2t dt

=%26gt; x=0 =%26gt; t = r and x = r =%26gt; t = 0

=%26gt; Volume

= 8π ∫ (t=r to t=0) [ - t2 ] dt

= 8π ∫ (t=0 to t=r) [ t2 ] dt

= (8π/3) t3 (t=0 to t=r)

= (8/3) π r3.


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