Finding Volume Using Cylindrical ShellsIt says use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis y = x 3 y = 8 about the axis x = 3. I drew the graph, reflected it about the x = 3 line, and drew a cylinder. `I figured that the radius is just r = 3 − x and the height would just be h = x 3 − 0 (since the lowest y value is zero), and plugged these into the integral of 2 ( π ) ( r ) ( h ) from 0 to 2. However, I got the wrong answer (correct answer should be 264 π / 5).I have a feeling that my height may be wrong but I'm not sure why.

Question

Finding Volume Using Cylindrical ShellsIt says use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis   y  =      x          3        y  =  8 about the axis   x  =  3. I drew the graph, reflected it about the   x  =  3 line, and drew a cylinder. `I figured that the radius is just   r  =  3  −  x and the height would just be   h  =      x          3        −  0 (since the lowest y value is zero), and plugged these into the integral of   2  (  π  )  (  r  )  (  h  ) from 0 to 2. However, I got the wrong answer (correct answer should be   264  π      /    5).I have a feeling that my height may be wrong but I&#039;m not sure why.

Kyle George · Accepted Answer

Explanation:Assuming the x and y intercepts are bounds in this problem (because you didn&#039;t mention it) the two equations intersect at   x  =  0 and   x  =  2, so those are the bounds of integration. By rotating around the line   x  =  3, the radius of the shell is   3  −  x and the height of the shell is   8  −      x    3  .Using the shell method, the integral would be set up like so:   2  π      ∫    0    2    (  3  −  x  )  (  8  −      x    3    )  d  x

Find the volume generated by rotating the region bounded by the given curves about the specified axis y=x^{3}\ y=8 about the axis x=3.

Answered question

Answer & Explanation

New Questions in High school geometry