Sketch the region bounded by the curves y=10√x and y=10x^3
 then set up the integral needed to find the volume of the solid generated by revolving this region about the x
-axis using the shell method.

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.



Sketch the region bounded by the curves y=10√x and y=10x^3 then set up the integral needed to find the volume of the solid generated by revolving this region about the x-axis using the shell method.

Learn how to find the volume of a solid formed by revolving the region bounded by the curves y=10√x and y=10x^3 about the x-axis using the shell method. This detailed guide includes curve intersections and integral setup, ideal for college-level calculus students.

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