Questions: Use the shell method to find the volume of the solid generated by revolving the shaded region about the x-axis. The volume is (Type an exact answer in terms of π).

Solution Steps

The region to be revolved is bounded by the curves \( y = \sqrt{5} \) and \( x = 15 - 3y^2 \), and the x-axis.

The shell method formula for revolving around the x-axis is: \[ V = 2\pi \int_{a}^{b} y \cdot \text{(radius)} \cdot \text{(height)} \, dy \]

• The limits of integration are from \( y = 0 \) to \( y = \sqrt{5} \).
• The radius is \( y \).
• The height is \( 15 - 3y^2 \).

\[ V = 2\pi \int_{0}^{\sqrt{5}} y (15 - 3y^2) \, dy \]

\[ V = 2\pi \int_{0}^{\sqrt{5}} (15y - 3y^3) \, dy \]

\[ V = 2\pi \left[ \frac{15y^2}{2} - \frac{3y^4}{4} \right]_{0}^{\sqrt{5}} \]

\[ V = 2\pi \left( \left[ \frac{15(\sqrt{5})^2}{2} - \frac{3(\sqrt{5})^4}{4} \right] - \left[ \frac{15(0)^2}{2} - \frac{3(0)^4}{4} \right] \right) \] \[ V = 2\pi \left( \left[ \frac{15 \cdot 5}{2} - \frac{3 \cdot 25}{4} \right] - 0 \right) \] \[ V = 2\pi \left( \left[ \frac{75}{2} - \frac{75}{4} \right] \right) \] \[ V = 2\pi \left( \frac{150}{4} - \frac{75}{4} \right) \] \[ V = 2\pi \left( \frac{75}{4} \right) \] \[ V = \frac{150\pi}{4} \] \[ V = \frac{75\pi}{2} \]

Final Answer