To solve this problem, let's break it down step by step.

1. Sketch the Region

The region is bounded by the two curves:

• $y = 10\sqrt{x}$
• $y = 10x^3$

These two curves intersect at points where $y$ values are equal for both equations. To find the points of intersection, set the two equations equal to each other:

$10\sqrt{x} = 10x^3$

Dividing both sides by 10:

$\sqrt{x} = x^3$

Square both sides to eliminate the square root:

$x = x^6$

Now, rearrange the equation:

$x^6 - x = 0$

Factor the equation:

$x(x^5 - 1) = 0$

This gives:

$x = 0 \quad \text{or} \quad x^5 = 1$

From $x^5 = 1$, we get:

$x = 1$

So, the curves intersect at $x = 0$ and $x = 1$.

2. Region Boundaries

The region is bounded by:

• $x = 0$
• $x = 1$
• $y = 10\sqrt{x}$ (above)
• $y = 10x^3$ (below)

3. Shell Method Setup for Volume of Solid of Revolution

The shell method is used to find the volume of the solid generated by revolving the region around the x-axis. In the shell method, the volume $V$ is given by the formula:

$V = \int_{a}^{b} 2\pi \cdot (\text{radius}) \cdot (\text{height}) \, dx$

Here:

• The "radius" is the distance from the axis of rotation (the x-axis), which is $y$.
• The "height" is the difference between the two functions (top minus bottom), which is $x$.

Now, revolving around the x-axis implies that the shells will be vertical with a height given by the x-values and radius by y-values.