SOLUTION: Find the volume of the solid generated by rotating the region enclosed by the curve 𝑦^2 = 16𝑥, the 𝑥 axis and the ordinate 𝑥 = 4 about: a) The 𝑥 axis b) The 𝑦 a

Question 1208484: Find the volume of the solid generated by rotating the region enclosed by the curve 𝑦^2 = 16𝑥, the 𝑥 axis and the ordinate 𝑥 = 4 about:
a) The 𝑥 axis
b) The 𝑦 axis
c) The line 𝑥 = 4
d) The line 𝑥 = 8

Answer by Shin123(626) (Show Source): You can put this solution on YOUR website!
We use the Washer Method for all of these.
a)
Note that we only consider the part above the x-axis. (the region below the x-axis gives the same results). This means that . This gives the volume as .
b)
We integrate with respect to this time, since we're rotating about the y-axis. Note that at , . (again, we're only considering the part above the x-axis) The outer radius is always , and the inner radius for a given is the corresponding x-value, which is . This means that the volume is .
c)
We also integrate with respect to . For a given , the radius is (no outer/inner radius this time), or . This means the volume is .
d)
We integrate with respect to . The inner radius is , and the outer radius for a given is , or . This means the volume is .
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