Volume by Shell and Washer Methods

[rodneyspencer]

1. The problem statement, all variables and given/known data

Find the volume generated by rotating the given region about the given line using the Shell method and the Washer method.

x = 4y [y = x/4], y = 0, x = 0, x = 8 about x

2. Relevant equations

Washer method (about x):
V = pi \(\displaystyle \int^b_a\) ((R[sub:17txrctt]top[/sub:17txrctt][sup:17txrctt]2[/sup:17txrctt](x) - r[sub:17txrctt]bottom[/sub:17txrctt][sup:17txrctt]2[/sup:17txrctt](x))dx

Shell method (about x):
V = 2pi \(\displaystyle \int^d_c\) (y[f(y)-g(y)])dy

3. The attempt at a solution

I'm not sure why I can't reconcile these two answers. I'm having some similar problems with more of these exercises but if someone can help me see where I'm going wrong I'm sure I can rework them successfully.

Washer:
V = pi \(\displaystyle \int^8_0\) ((x/4)[sup:17txrctt]2[/sup:17txrctt])dx = 32pi/3

Shell:
V = 2pi \(\displaystyle \int^2_0\) (y(4y))dy = 64pi/3

 

[tkhunny]

Excellent work. You are SO CLOSE! Too bad you are wrapping the wrong piece in the Shells. Try 8-4y, rather than 4y.

 

[rodneyspencer]

Ahh, because the shell is with respect to the y-axis and x=8 is my second function! No wonder! Thank you so much