We are asked to find the volume of the solid obtained by rotating the region bounded by $y = 27x^3$, $y = 0$, and $x = 1$, about the line $x = 2$. This requires using the method of cylindrical shells because we are rotating about a vertical line $x = 2$, and our region is described in terms of $x$.

Step 1: Setup of the cylindrical shells formula

For a solid obtained by rotating around a vertical line, the volume $V$ using cylindrical shells is given by the formula: $V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx$ Here:

• The radius is the distance from the curve to the axis of rotation. In this case, the distance from $x$ to $x = 2$ is $2 - x$.
• The height is the function value, which is $y = 27x^3$.

The limits of integration will be from $x = 0$ to $x = 1$ because these are the bounds of the region.

Step 2: Plugging in the values

Now we can set up the integral for the volume: $V = 2\pi \int_0^1 (2 - x) \cdot 27x^3 \, dx$ Simplifying the integrand:

= 2\pi \int_0^1 (54x^3 - 27x^4) \, dx$$ ### Step 3: Solve the integral Now we compute the integral term by term: $$V = 2\pi \left[ \int_0^1 54x^3 \, dx - \int_0^1 27x^4 \, dx \right]$$ First, calculate the integrals: $$\int_0^1 54x^3 \, dx = 54 \left[ \frac{x^4}{4} \right]_0^1 = 54 \times \frac{1}{4} = 13.5$$ $$\int_0^1 27x^4 \, dx = 27 \left[ \frac{x^5}{5} \right]_0^1 = 27 \times \frac{1}{5} = 5.4$$ ### Step 4: Substitute and simplify Now substitute these values back into the expression for $$V$$: $$V = 2\pi \left( 13.5 - 5.4 \right) = 2\pi \times 8.1 = 16.2\pi$$ Thus, the volume of the solid is: $$V = 16.2\pi \, \text{cubic units}$$ Would you like more details on any of the steps or the method used? Here are some related questions to extend your understanding: 1. What would the volume be if the region was rotated about the y-axis instead of $$x = 2$$? 2. How would the volume change if the function was $$y = 27x^2$$ instead of $$y = 27x^3$$? 3. What happens to the volume if the bounds of $$x$$ are changed from $$0 \leq x \leq 1$$ to $$0 \leq x \leq 2$$? 4. Can you use the disk method to solve this problem? Why or why not? 5. What is the volume if the rotation axis was moved to $$x = -1$$ instead? **Tip**: For rotating regions around a vertical line that isn't the y-axis, the cylindrical shells method is often the best approach!