To find the volume of the solid generated when the area enclosed by the curves \( y = 0 \), \( y = 2x \), and \( x = 3 \) is rotated about the x-axis, we will use the method of disks (or washers). Let's go through the steps in detail. ### Step 1: Understand the Area Enclosed First, we need to identify the area that is being rotated. The curves given are: - \( y = 0 \) (the x-axis) - \( y = 2x \) (a straight line) - \( x = 3 \) (a vertical line) To find the points of intersection, we can set \( y = 2x \) equal to \( y = 0 \): - Setting \( 2x = 0 \) gives \( x = 0 \). Thus, the area is bounded by: - The x-axis from \( x = 0 \) to \( x = 3 \) - The line \( y = 2x \) from \( x = 0 \) to \( x = 3 \) ### Step 2: Set Up the Volume Integral When we rotate this area around the x-axis, we can visualize it as a series of disks stacked along the x-axis. The radius of each disk at a point \( x \) is given by the function \( y = 2x \). The volume \( V \) of the solid of revolution can be calculated using the formula: \[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \] where \( f(x) \) is the function representing the radius of the disks, and \( [a, b] \) is the interval along the x-axis. In our case: - \( f(x) = 2x \) - The limits of integration are from \( x = 0 \) to \( x = 3 \). ### Step 3: Calculate the Volume Now we can set up the integral: \[ V = \pi \int_{0}^{3} (2x)^2 \, dx \] Calculating \( (2x)^2 \): \[ (2x)^2 = 4x^2 \] Now substitute this back into the integral: \[ V = \pi \int_{0}^{3} 4x^2 \, dx \] Next, we can factor out the constant: \[ V = 4\pi \int_{0}^{3} x^2 \, dx \] Now we need to evaluate the integral \( \int x^2 \, dx \): \[ \int x^2 \, dx = \frac{x^3}{3} \] Now we can evaluate this from \( 0 \) to \( 3 \): \[ \int_{0}^{3} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{0}^{3} = \frac{3^3}{3} - \frac{0^3}{3} = \frac{27}{3} - 0 = 9 \] Now substitute this back into the volume formula: \[ V = 4\pi \cdot 9 = 36\pi \] ### Conclusion Thus, the volume of the solid generated when the area enclosed by the curves is rotated about the x-axis is: \[ \boxed{36\pi} \text{ cubic units} \] ### Explanation of Other Options - **Option A (81 π cubic units)**: This option is incorrect because it suggests a volume that is too large. The calculations show that the volume is significantly less than this. - **Option C (18 π cubic units)**: This option is also incorrect. It underestimates the volume, likely due to an error in the integration or misunderstanding of the area being rotated. - **Option D (9 π cubic units)**: This option is incorrect as well. It is too small and does not account for the correct limits and the function being squared. ### Revision Summary - The volume of a solid of revolution can be calculated using the disk method. - The formula for volume is \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \). - For the given curves, the correct integral setup leads to \( V = 36\pi \) cubic units. - Always check the limits of integration and the function being squared to avoid common pitfalls.