Question

Compute the volume of the solid obtained by rotating the region in the first quadrant encolsed by the graphs of the functions y = x^2 and y = sqrt(x) about the y-axis.
I keep getting (1/3)pi and can't figure out what I'm doing wrong!

Answer 1

rotating about the y -axis needs
V = π∫x^2 dy

first equation:
y = x^2
2nd equation:
√x = y
x = y^2
x^2 = y^4

so V = π∫y dy - π∫y^4 dy
= π [ (1/2)y^2 - (1/5)y^5] from y = 0 to 1
= π( 1/2 - 1/5)
= (3/10)π