Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

y=4 x^2, x = 1, y = 0, about the x-axis

Is there a lower x limit?

If there isn't, the volume is infinite

is the region bounded by x=0? if no, then it is in fact infinite...

if yes, then:

1) find the points of intersection. they are x = 1/2 and x = 0.
2) take the integral of pi*(4x^2)^2 from 0 to 1/2.

To find the volume of the solid obtained by rotating the region bounded by the curves y = 4x^2, x = 1, and y = 0 about the x-axis, we can use the disk method.

The disk method involves integrating the cross-sectional areas of infinitesimally thin disks that make up the solid of revolution.

Let's break down the problem step-by-step:

1. Sketch the region bounded by the curves:
- The curve y = 4x^2 is a symmetric parabola opening upwards and crossing the x-axis at x = 0.
- The line x = 1 is a vertical line passing through x = 1.
- The line y = 0 is the x-axis.

Sketching the region will help us visualize the solid and its rotational axis.

2. Determine the limits of integration:
- Since we are rotating about the x-axis, the limits of integration will be x-values that define the region.
- In this case, the region is bounded by x = 0 and x = 1.

3. Set up the integral for the volume:
- The volume of each infinitesimally thin disk is given by π * (radius)^2 * (height).
- In this case, the radius of each disk is the y-coordinate of the curve y = 4x^2, which is 4x^2.
- The height of each disk is an infinitesimally small change in x, represented by dx.
- Thus, the volume of each disk is given by dV = π * (4x^2)^2 * dx.
- To find the total volume, we will integrate this expression from x = 0 to x = 1.
- The integral for the volume is ∫[0,1] π * (4x^2)^2 * dx.

4. Calculate the volume:
- Evaluating the integral will give us the volume of the solid.
- ∫[0,1] π * (4x^2)^2 * dx = π * ∫[0,1] 16x^4 * dx.

To calculate this integral, we can use the power rule of integration:
- Applying the power rule, we get π * (16/5) * [x^5] from 0 to 1.
- Evaluating the expression at the limits of integration gives:
V = π * (16/5) * [(1)^5 - (0)^5]
= π * (16/5) * 1
= 16π/5 or (16/5)π.

So, the volume of the solid obtained is (16/5)π cubic units or approximately 10.1 cubic units.