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

Algebra ->  Quadratic-relations-and-conic-sections -> 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      Log On


   

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) About Me  (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 y=4%5Csqrt%28x%29. This gives the volume as .
b)
We integrate with respect to y this time, since we're rotating about the y-axis. Note that at x=4, y=8. (again, we're only considering the part above the x-axis) The outer radius is always 4, and the inner radius for a given y is the corresponding x-value, which is y%5E2%2F16. This means that the volume is .
c)
We also integrate with respect to y. For a given y, the radius is 4-x (no outer/inner radius this time), or 4-y%5E2%2F16. This means the volume is .
d)
We integrate with respect to y. The inner radius is 4, and the outer radius for a given y is 8-x, or 8-y%5E2%2F16. This means the volume is .