Find the volume?

Question

Suppose a is a constant between 0 and 1
Let R be the region between y=ax-x^2 and the x-axis from x=0 to x=1.

a) What is the volume of the solid obtained by rotating R about the x-axis?

b) What value of a creates the solid with the smallest volume?

Answer 1

Hello,

The formula for the volume for a shell is the integral of the circumference * the height.. So we have integral form 0 to 1 of 2pi*x*y dx the we have integral from 0 to 1 of 2pi*x*(ax - x^2) dx then taking the integral we have.

2pi*[(1/3)*ax^3)) -[(1/4)*x^4]] from 0 to 1

Of course at 0 everything = 0 so we have at 1

2pi*[1/3a -1/4] = Volume

Now to get the min we set to = 0 and we get

2pi*[1/3a - 1/4] - so

1/3a = 1/4 then a = 3/4


Hope This Helps!!

Answer 2

use integration formula