integration volume

Sub. x^2 = y^3 into y = 10 - x^2
y = 10 - y^3
y^3 + y - 10 = 0
(y - 2)(y^2 + 2y + 5) = 0
y = 2
x = [8^(1/2)] or -[8^(1/2)]
thus, the intersection points are (8^(1/2), 2) and (-8^(1/2), 2)

therefore, the volume
= { [pi(10 - x^2)^2 - pi(x^(2/3))^2] dx
= pi { [100 - 20x^2 + x^4 - x^(4/3)] dx
= pi [100x - 20/3 x^3 + 1/5 x^5 - 3/7 x^(7/3)]
= pi [100x - 20/3 [8^(1/2)]^3 + 1/5 [8^(1/2)]^5 - 3/7 [8^(1/2)]^(7/3)]
- pi [100x - 20/3 [-8^(1/2)]^3 + 1/5 [-8^(1/2)]^5 - 3/7 [-8^(1/2)]^(7/3)]
= pi [200[8^(1/2)] - 40/3 [8^(1/2)]^3 + 2/5 [8^(1/2)]^5 - 6/7 [8^(1/2)]^(7/3)]
= 1026.3