Question

Which of the following integrals correctly computes the volume formed when the region bounded by the curves x^2 + y^2 = 25, x = 4 and y = 0 is rotated around the y-axis?

A. pi ∫ upper bound of 3 and lower bound of 0 ( sqrt(25 - y^2) -4)^2 dy

B. pi ∫ upper bound of 3 and lower bound of 0 ( 4^2 - ( sqrt(25-y^2) )^2 ) dy

C. pi ∫ upper bound of 5 and lower bound of 4 ( sqrt(25-x^2) )^2 dy

D. pi ∫ upper bound of 3 and lower bound of 0 ( (sqrt(25-y^2)^2 ) - 4^2 ) dy

I'm struggling on this, please helpp!
Thank you so so much in advance!

Answer 1

The region being rotated is the triangular slice with vertices at (4,0), (4,3), (5,0). Recall that for washers, the volume is
v = ∫ π(R^2-r^2) dy
Here, R is the circle, and r is the line x=4. So, choice (D)

Answer 2

choice d is right