Calculus Problem

Question

Find the volume of the solid obtained by rotating the region bounded by the curves y= x²/ 4, x=0 and y=4 about the y-axis

Answer 1

PROBLEM

 

Calculus Problem
Find the volume of the solid obtained by rotating the region bounded by the curves y=x²/4, x=0 and y=4 about the y-axis

 

SOLUTION

 

Since the volume is created by revolving about the y-axis, you can integrate with respect to y...

 

So, use the formula (disc method) V = ∫^b_a π f(y)² dy

 

Rewrite the equations in terms of y...

 

y=x²/4

 

4y=x²

 

√4y=√(x²)

 

2√y=x

 

x=2√y (this will be rightmost radius)

 

x=0 (this will be leftmost radius)

 

Integrate from y=0 to y=4

 

V = ∫⁴₀ π (2√y)² dy

 

V = ∫⁴₀ π (4y) dy

 

V = π (2y²)|⁴₀

 

V = π [2(4)² - 0]

 

V = 32π