Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = 27x^3,  y = 0,  x = 1;  about x = 2

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:
\[
V = 2\pi \int_0^1 27x^3(2 - x) \, dx
= 2\pi \int_0^1 (54x^3 - 27x^4) \, dx
\]

### 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!

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line: y = 27x^3, y = 0, x = 1; about x = 2.

Learn how to calculate the volume of a solid obtained by rotating the region bounded by y = 27x^3, y = 0, and x = 1 around x = 2. This problem demonstrates the cylindrical shells method and involves integration techniques. Perfect for students studying advanced calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation