Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8.
27y = x3, y = 0, x = 6

Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about y = 8.
27y = x3,    y = 0,    x = 6

tkhunny · Answer

Super!  Let's see your first attempts.  Have you identified the region?

anonymous · Answer

i cant figure out where to start. i know that v= integral of a to b 2 pi x f(x) dx

tkhunny · Answer

You must first get a handle on the region.  What are the limits of the surface?

x is in [0,6]
y is in [0,8]

Are we talking about the portion below the curve and adjacent to the x-axis or the portion above the curve and adjacent to the y-axis?

Interestingly, $\dfrac{1}{27}\cdot 6^{3} = 2^{3} = 8$  It appears the distant point of the integration is (6,8)

anonymous · Answer

so for starters, would it be the integral from 6 to 8 =2pi x * x^3/27 dx?

tkhunny · Answer

That makes no sense. 6 is from x and 8 is from y. Why would they be in the limits of the same integral? Give it a better effort.

tkhunny · Answer

You are wrapping around y = 8.

What is the radius of any circle with center at y = 8?  Hopefully, it is 8-y.

What is the height of any such section?  Well, since we have not yet established which of the two pieces were dealing with, we can't answer that, yet.  Which piece are we dealing with?  Adjacent to x-axis or adjacent to y-axis?  You MUST know or you cannot solve this problem.