Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = x2,   y = 3x;
   about the y-axis

We are asked to find the volume of the solid obtained by rotating the region bounded by the curves \( y = x^2 \) and \( y = 3x \) about the y-axis.

### Step 1: Find the points of intersection
To determine the region bounded by the curves, we first find the points of intersection by setting the two equations equal:

\[
x^2 = 3x
\]

Solve for \( x \):

\[
x^2 - 3x = 0
\]

\[
x(x - 3) = 0
\]

Thus, \( x = 0 \) and \( x = 3 \).

### Step 2: Set up the integral using the method of cylindrical shells
Since we are rotating the region about the **y-axis**, we use the method of cylindrical shells. The formula for the volume of a solid of revolution using shells is:

\[
V = 2\pi \int_{a}^{b} x \left( f(x) - g(x) \right) \, dx
\]

where \( f(x) \) is the upper curve and \( g(x) \) is the lower curve. Here, the upper curve is \( y = 3x \) and the lower curve is \( y = x^2 \), so we have:

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

### Step 3: Simplify the integrand
Simplify the integrand:

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

### Step 4: Integrate
Now, we compute the integral:

\[
V = 2\pi \left[ \int_{0}^{3} 3x^2 \, dx - \int_{0}^{3} x^3 \, dx \right]
\]

The integrals are straightforward:

\[
\int 3x^2 \, dx = x^3
\quad \text{and} \quad
\int x^3 \, dx = \frac{x^4}{4}
\]

So,

\[
V = 2\pi \left[ x^3 - \frac{x^4}{4} \right]_{0}^{3}
\]

### Step 5: Evaluate the definite integrals
Now, evaluate at the limits \( x = 0 \) and \( x = 3 \):

\[
V = 2\pi \left[ \left( 3^3 - \frac{3^4}{4} \right) - \left( 0^3 - \frac{0^4}{4} \right) \right]
\]

Simplify:

\[
V = 2\pi \left( 27 - \frac{81}{4} \right)
= 2\pi \left( \frac{108}{4} - \frac{81}{4} \right)
= 2\pi \left( \frac{27}{4} \right)
\]

\[
V = \frac{54\pi}{4} = \frac{27\pi}{2}
\]

### Final Answer:
The volume of the solid is:

\[
V = \frac{27\pi}{2} \, \text{cubic units}
\]

Would you like any further details or explanations?

Here are 5 related questions to expand on this concept:
1. How would the volume change if we rotated the region around the x-axis instead of the y-axis?
2. Can you explain how to set up the integral using the disk method for this problem?
3. What would the integral setup look like if we were rotating around the line \( x = 1 \)?
4. How would you compute the volume if the region was bounded by \( y = x^2 \) and \( y = 2x \) instead?
5. Can you derive the formula for the method of cylindrical shells?

**Tip**: When using cylindrical shells, always ensure the radius and height of the shells are expressed in terms of the axis of rotation for accuracy.

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

Learn how to compute the volume of the solid formed by rotating the region bounded by y = x^2 and y = 3x around the y-axis using the method of cylindrical shells. Detailed steps include finding the points of intersection, setting up the integral, and solving. Suitable for Grades 11-12 calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation