Expert: Ahmed Salami - 12/4/2009

Question
find the volume of the solid generated by revolving the region bounded by x =2y^2, y =x, about the y-axis.

Answer
Hi Andrew,
First we need to determine where the lines intersect.
x = 2y² and x = y
They intersect at
2y² = y
2y² - y = 0
y(2y - 1) = 0
y = 0 or 2y - 1 = 0
y = 0 or 1/2
Now, let x1 = 2y² and x2 = y
The volume generated by revolving about the y-axis would be
∫π(x2)² dy - ∫π(x1)² dy             (between the limits y = 0 and 1/2)
= ∫π[(x2)² - (x1)²] dy
= ∫π[y² - (2y²)²] dy
= ∫π(y² - 4y^4) dy
= π(y^3/3 - 4y^5/5)   from 0 to 1/2
= π[(1/2)^3/3 - 4(1/2)^5/5] - π[(0)^3/3 - 4(0)^5/5]
= π[(1/3)(1/8) - (4/5)(1/32)]
= π[(1/24) - (1/40)]
= π(1/60)
= π/60 cubic units

Regards