To find the volume of the solid generated by revolving the shaded region about the x-axis, we use the disk method. The volume of a solid of revolution using the disk method is given by:

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

Step 1: Identify the function and limits

The given function is:

$y = 6 - 5x$

The region is bounded by:

• $y = 6 - 5x$ (upper boundary)
• $y = 0$ (x-axis)
• $x = 0$ (left boundary)
• $x = \frac{6}{5}$ (right boundary)

Step 2: Set up the integral

Since we are revolving around the x-axis, we use the disk method formula:

$V = \pi \int_{0}^{\frac{6}{5}} (6 - 5x)^2 \, dx$

Thus, the required integral is:

$V = \pi \int_{0}^{\frac{6}{5}} (6 - 5x)^2 \, dx$

Would you like a step-by-step solution for evaluating the integral?

Related Questions

1. How do we use the shell method to compute this volume instead?
2. What if we revolved the region around the y-axis instead of the x-axis?
3. How does the volume change if we used the washer method instead?
4. What happens if we integrate in terms of y instead of x?
5. How do we verify the result using numerical integration?

Tip:

When setting up volume integrals, always check whether disks, washers, or shells are the best method based on the axis of rotation!