Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the​ x-axis.
yequals5 x​,  yequals5​,  xequals0

To find the volume of the solid generated by revolving the region bounded by the curve \( y = 5x \), the line \( y = 5 \), and the line \( x = 0 \) around the \( x \)-axis, we will use the **disk method**.

### Steps:

1. **Set up the bounds:**
   The region is bounded by \( y = 5x \), the horizontal line \( y = 5 \), and the vertical line \( x = 0 \).

   We need to find the limits of integration by determining where the curve intersects the line \( y = 5 \):
   \[
   y = 5x \quad \text{and} \quad y = 5.
   \]
   Setting these equal:
   \[
   5x = 5 \quad \Rightarrow \quad x = 1.
   \]
   So the region is bounded by \( x = 0 \) and \( x = 1 \).

2. **Disk method formula:**
   The volume of the solid generated by revolving around the \( x \)-axis can be found using:
   \[
   V = \pi \int_{a}^{b} [f(x)]^2 \, dx.
   \]
   Here, \( f(x) = 5x \), and the limits of integration are from \( x = 0 \) to \( x = 1 \).

3. **Set up the integral:**
   \[
   V = \pi \int_{0}^{1} (5x)^2 \, dx.
   \]
   Simplifying the integrand:
   \[
   V = \pi \int_{0}^{1} 25x^2 \, dx.
   \]

4. **Evaluate the integral:**
   \[
   V = \pi \left[ \frac{25x^3}{3} \right]_{0}^{1}
   \]
   Substituting the limits:
   \[
   V = \pi \left( \frac{25(1)^3}{3} - \frac{25(0)^3}{3} \right)
   \]
   \[
   V = \pi \times \frac{25}{3} = \frac{25\pi}{3}.
   \]

Thus, the volume of the solid is \( \frac{25\pi}{3} \) cubic units.

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Would you like more details or have any questions?

### Related questions:
1. How does the shell method differ from the disk method for finding volumes?
2. How would you find the volume if the region was revolved around the y-axis instead of the x-axis?
3. What happens if we revolve a region bounded by a different curve, such as \( y = x^2 \)?
4. How would the integral change if the limits were different?
5. Can the disk method be applied to regions not bounded by a line parallel to the axis of revolution?

**Tip:** Always sketch the region and the axis of rotation to better visualize the problem.

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the​ x-axis. y = 5x, y = 5, x = 0

Learn how to calculate the volume of a solid generated by revolving the region bounded by y = 5x, y = 5, and x = 0 around the x-axis using the disk method. This problem is solved step-by-step, focusing on integral calculus concepts. Ideal for Grades 11-12 or introductory calculus students.

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