Find the volume of the solid obtained by rotating the region bounded by
y=4x^2,x=1,y=0
about the x-axis.
To find the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, y = 0 about the x-axis, we can use the method of cylindrical shells.
1. First, let's visualize the region bounded by these curves:
The curve y = 4x^2 is a parabola that opens up, with its vertex at the origin. The line x = 1 is a vertical line passing through the point (1, 0). The line y = 0 is the x-axis.
So, the region we need to rotate is a parabolic segment between the x-axis and the vertical line x = 1.
2. Next, let's determine the limits of integration:
To find the limits of integration, we need to find the x-values where the parabolic segment intersects the vertical line x = 1.
Setting y = 4x^2 equal to 0 and solving for x, we get:
4x^2 = 0
x = 0
So, the region of interest lies between x = 0 and x = 1.
3. Now, let's set up the integral for the volume using the method of cylindrical shells:
The volume of a cylindrical shell can be calculated using the formula:
dV = 2πrh ∆x
where r is the radius, h is the height, and ∆x is the thickness of the shell.
In this case, the radius is x because the shell is formed by rotating the region about the x-axis.
The height is y = 4x^2.
The thickness of the shell is ∆x.
The integral for the volume can be written as:
V = ∫(2πx)(4x^2)dx
= ∫8πx^3 dx
4. Evaluate the integral:
Integrating 8πx^3 with respect to x, we get:
V = 2π[x^4] from 0 to 1
= 2π(1^4 - 0^4)
= 2π(1)
= 2π
Therefore, the volume of the solid obtained by rotating the region bounded by y = 4x^2, x = 1, y = 0 about the x-axis is 2π cubic units.