Question

Find the volume of the solid obtained by rotating the region bounded by the curves
y=x^4,y=1
about the line y=4 .

Answer:

Answer 1

intersection of y = x^4 and y = 4
y^4 = 4
y^2 = ±2
y = ± √2
intersection of y = x^4 and y = 1 -----> x = ±1

Nice symmetry, so instead of ±√2, let's go from 0 to √2, then double our answer.

Volume = 2π∫ ( (4 - x^4)^2 - (1 - x^4)^2 )dx from 0 to √2
= 2π ∫ (15 - 6x^4) dx from 0 to √2
= .....
routine from here