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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