Questions: Find the volume of the solid obtained by rotating the region bounded by the curve f(x) = e^(3x)/6, the x-axis, the y-axis, and the line x=1 around the y-axis. Submit an exact answer. Provide your answer below: Volume = (π/27)(2e^3+1)

Solution Steps

To find the volume of the solid obtained by rotating the given region around the y-axis, we can use the method of cylindrical shells. The formula for the volume using cylindrical shells is:

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

In this problem, the region is bounded by \( f(x) = \frac{e^{3x}}{6} \), the x-axis, the y-axis, and the line \( x = 1 \). We will integrate from \( x = 0 \) to \( x = 1 \).

1. Set up the integral for the volume using the method of cylindrical shells.
2. Integrate the function from \( x = 0 \) to \( x = 1 \).
3. Evaluate the integral to find the exact volume.

To find the volume of the solid obtained by rotating the region around the \( y \)-axis, we use the method of cylindrical shells. The formula for the volume is:

\[ V = 2\pi \int_{0}^{1} x \cdot \frac{e^{3x}}{6} \, dx \]

We evaluate the integral:

\[ V = 2\pi \left( \int_{0}^{1} \frac{x e^{3x}}{6} \, dx \right) \]

The integral evaluates to:

\[ \int_{0}^{1} \frac{x e^{3x}}{6} \, dx = \frac{1}{54} + \frac{e^{3}}{27} \]

Substitute the evaluated integral back into the volume formula:

\[ V = 2\pi \left( \frac{1}{54} + \frac{e^{3}}{27} \right) \]

This simplifies to approximately:

\[ V \approx 4.790 \]

Final Answer