https://file.aiquickdraw.com/mathbot/file/1725912095301428bceww.jpg

The problem provided asks to find the volume of a solid obtained by rotating the region under the curve \( y = x\sqrt{4 - x^2} \) from \( x = 0 \) to \( x = 2 \) about the y-axis. This requires applying the method of cylindrical shells.

### Step-by-step solution:

1. **Formula for volume using cylindrical shells:**

   When rotating around the y-axis, the volume \( V \) is given by the formula:
   \[
   V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
   \]
   where \( f(x) \) is the function describing the curve, and \( a \) and \( b \) are the limits of integration (in this case, \( 0 \) and \( 2 \)).

2. **Identify the function and limits:**
   - \( f(x) = x\sqrt{4 - x^2} \)
   - The limits are from \( x = 0 \) to \( x = 2 \).

3. **Set up the integral:**
   \[
   V = 2\pi \int_{0}^{2} x \cdot x\sqrt{4 - x^2} \, dx = 2\pi \int_{0}^{2} x^2 \sqrt{4 - x^2} \, dx
   \]

4. **Substitute \( u = 4 - x^2 \):**
   - \( du = -2x \, dx \)
   - When \( x = 0 \), \( u = 4 \).
   - When \( x = 2 \), \( u = 0 \).

   The integral becomes:
   \[
   V = 2\pi \int_{4}^{0} -\frac{1}{2} \sqrt{u} \, (-du) = \pi \int_{0}^{4} u^{1/2} \, du
   \]

5. **Integrate:**
   \[
   \pi \int_{0}^{4} u^{1/2} \, du = \pi \left[ \frac{2}{3} u^{3/2} \right]_{0}^{4}
   \]
   \[
   V = \pi \left( \frac{2}{3} (4)^{3/2} - \frac{2}{3} (0)^{3/2} \right) = \pi \left( \frac{2}{3} \cdot 8 \right) = \frac{16\pi}{3}
   \]

So, the volume of the solid is \( \frac{16\pi}{3} \) cubic units.

---

Would you like further details or have any questions?

Here are some related problems:
1. What is the volume when rotating the same region around the x-axis?
2. How would you solve this problem if the curve were rotated around the line \( x = 1 \)?
3. Can you calculate the surface area of the solid obtained by this rotation?
4. How does the volume change if the curve is rotated from \( x = -2 \) to \( x = 2 \)?
5. What is the interpretation of the integral when rotating a curve about the y-axis?

**Tip:** When using the cylindrical shells method, always ensure that you set up the integral correctly in terms of the radius and height of the shell.

Find the volume of the solid obtained when the region under the curve y = x√(4 - x²) from x = 0 to x = 2 is rotated about the y-axis.

Learn how to calculate the volume of the solid obtained by rotating the region under the curve y = x√(4 - x²) from x = 0 to x = 2 about the y-axis. This problem uses the method of cylindrical shells and is ideal for students studying Calculus II.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation