Find the volume of the solid that results by revolving the region enclosed by the curves y=5x, x=1, x=2, and y = 0 about the x-axis

We are asked to find the volume of the solid formed by revolving the region enclosed by the curves \( y = 5x \), \( x = 1 \), \( x = 2 \), and \( y = 0 \) about the x-axis.

### Step 1: Determine the method
Since the region is being revolved about the x-axis, we can use the **disk method**. The formula for the volume of a solid of revolution using the disk method is:

\[
V = \pi \int_{a}^{b} \left[ f(x) \right]^2 dx
\]

Where:
- \( f(x) \) is the function being revolved,
- \( a \) and \( b \) are the bounds of the region along the x-axis.

### Step 2: Set up the integral
The function \( f(x) = 5x \), and the limits of integration are \( x = 1 \) and \( x = 2 \). The volume formula becomes:

\[
V = \pi \int_{1}^{2} (5x)^2 dx
\]

### Step 3: Simplify the integrand
First, simplify \( (5x)^2 \):

\[
(5x)^2 = 25x^2
\]

Now the volume integral becomes:

\[
V = \pi \int_{1}^{2} 25x^2 dx
\]

### Step 4: Compute the integral
Now, we compute the integral of \( 25x^2 \):

\[
V = \pi \times 25 \int_{1}^{2} x^2 dx
\]

The integral of \( x^2 \) is:

\[
\int x^2 dx = \frac{x^3}{3}
\]

Now, apply the limits of integration:

\[
V = 25\pi \left[ \frac{x^3}{3} \right]_{1}^{2}
\]

### Step 5: Evaluate the limits
Substitute \( x = 2 \) and \( x = 1 \) into \( \frac{x^3}{3} \):

\[
V = 25\pi \left( \frac{2^3}{3} - \frac{1^3}{3} \right)
\]

\[
V = 25\pi \left( \frac{8}{3} - \frac{1}{3} \right)
\]

\[
V = 25\pi \times \frac{7}{3}
\]

\[
V = \frac{175\pi}{3}
\]

### Final Answer:
The volume of the solid is \( \frac{175\pi}{3} \) cubic units.

Would you like any further details or have any questions?

Here are some related questions for practice:
1. What is the volume if the region is revolved around the line \( y = -2 \)?
2. How would the method change if we revolved the region around the y-axis?
3. What would be the volume if the function was \( y = 5x^2 \) instead of \( y = 5x \)?
4. How does the washer method differ from the disk method for similar problems?
5. Can the volume be computed using the shell method, and if so, how?

**Tip:** Always sketch the region and axis of revolution to get a clear picture of the problem before applying any methods.

Find the volume of the solid that results by revolving the region enclosed by the curves y=5x, x=1, x=2, and y=0 about the x-axis

This detailed solution explains how to find the volume of a solid formed by revolving the region enclosed by y=5x, x=1, x=2, and y=0 about the x-axis. Using the disk method, we compute the integral step-by-step, providing a clear understanding of the process. Ideal for college-level calculus students.

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