Calculus Problem...?

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

y = (1/x^5), y = 0, x = 4, x = 5;

about the y-axis.

Answer 1

For y between 1/5^5 and 1/4^5, the x values extend from 4 to y^(-1/5). For y between 0 and 1/5^5, the x values extend from 4 to 5.

So we have
π∫(0 to 1/5^5) (5^2 - 4^2) dy + π∫(1/5^5 to 1/4^5) (y^(-2/5) - 4^2) dy
= 9π/5^5 + π[y^(3/5) / (3/5)][1/5^5 to 1/4^5] + π(-16)(1/4^5 - 1/5^5)
= π(9/5^5 + (5/3)[4^(-3) - 5^(-3)] - 1/4^3 + 16/5^5)
= π(1/5^3 + (5/3)(1/4^3) - (5/3)(1/5^3) - 1/4^3)
= π(2/3)(1/4^3 - 1/5^3)
= 61π/12000.