Let A be the bounded region enclosed by f(x) = x ; g(x) = x4 :?

Question

Find the volume of the solid obtained by ro-
tating the region A about the line x + 4 = 0 :

Answer 1

y = x and y = x^4 intersect only at (0, 0) and (1, 1). On [0, 1] both functions are monotonic increasing. For a given value of y in [0, 1], x = y will be smaller than x = y^(1/4).

So the integral becomes
∫(y=0 to 1) ((y^(1/4))^2 - y^2) dy
= ∫(y=0 to 1) (y^(1/2) - y^2) dy
= [y^(3/2) / (3/2) - y^3 / 3] [y=0 to 1]
= (1 / (3/2) - 1/3) - (0 / (3/2) - 0/3)
= 2/3 - 1/3 = 1/3.