How to find the integral of a solid using disks or washer's method when rotating the region enclosed by y=1/x^5, y=0, x=3, x=7 about the line y=-5?

Question

The volume of the solid obtained by rotating the region enclosed by y= 1/x⁵, y= 0, x= 3, x= 7 about the line y= -5.

Find the integral (disks or washer's method) needed.

V= ∫ ... dx

I know that a= 3 & b= 7 and that it's about the x-axis, but I just can't seem to find the integral.

Answer 1

Hello,

You mentioned I know that a= 3 & b= 7 and that it's about the x-axis, but I just can't seem to find the integral, which is not correct.

It is y = -5.

Hence, given the area and revolving axis, there is space between them. We need to use the Washer method.

I will say the bigger circle's radius is R, and the smaller circle's radius is r.

R = f(x) + 5 =====> rectangle height + distance between x-axis and y = - 5

r = 5 ===========> distance between x-axis and y = - 5

Integra (3,7) of pi * ( R^2 - r^2 ) dx = integral (3,7) of pi * ((x^-5) ^ 2 + 10 * x^(-5) + 25 - 25) dx

= pi S (3,7) of ((x^-5) ^ 2 + 10 * x^(-5)) dx = pi [ - 1/9 * x^(-9) - 10/4 * x^(-4)] from 3 to 7

= pi [ -1/9 * (7^ (-9) - 3 ^ (-9)) - 5/2 ( 7^(-4) - 3^(-4))] = 0.09370933258

Around .09371

Answer 2

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