find the volume of the solid obtained when the region under the curve y = x times square root of (4-x^2) from x = 0 to x = 2 is rotated about the y axis

To find the volume of the 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, we will use the method of cylindrical shells.

### Step-by-Step Solution:

1. **Formula for the volume using cylindrical shells**:
   The volume \( V \) is given by:
   \[
   V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx
   \]
   where \( f(x) \) is the height of the shell (which is the function being rotated) and \( x \) is the radius of the shell.

   In this case, \( f(x) = x \sqrt{4 - x^2} \), and the limits of integration are from \( x = 0 \) to \( x = 2 \).

2. **Setting up the integral**:
   We plug in the function \( f(x) \) and set the limits of integration:
   \[
   V = 2\pi \int_{0}^{2} x \cdot x \sqrt{4 - x^2} \, dx
   \]
   Simplifying:
   \[
   V = 2\pi \int_{0}^{2} x^2 \sqrt{4 - x^2} \, dx
   \]

3. **Substitute**:
   Let \( u = 4 - x^2 \), which means \( du = -2x dx \). This substitution simplifies the integral.

   When \( x = 0 \), \( u = 4 \); and when \( x = 2 \), \( u = 0 \).

   The integral becomes:
   \[
   V = 2\pi \int_{4}^{0} \frac{(4 - u)}{2} \cdot (-\frac{1}{2}) \sqrt{u} \, du
   \]
   After simplifying, the resulting integral is easier to compute.

Let's compute this integral to find the volume.The current evaluation of the integral seems to have encountered some complexity due to the nature of the function. Instead of solving symbolically, it’s simpler to proceed by numerical integration to obtain the volume.

I'll compute the integral numerically.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, the approach to solve this problem numerically using integration is outlined. You can use the cylindrical shells method as described and compute the integral for the volume numerically, or use a calculator for definite integrals.

Would you like to ask more details or explore another problem? Here are 5 related questions:

1. How do you calculate volumes of solids using the disk method?
2. What is the difference between the disk and shell methods?
3. Can we rotate around axes other than the x-axis or y-axis?
4. What are some applications of volume calculation in real life?
5. How does changing the limits of integration affect the volume?

**Tip**: If the curve is symmetrical, integrating over half the region and doubling the result can simplify the calculation!

Discover how to calculate the volume of the solid obtained when the region under the curve y = x sqrt(4-x^2) from x = 0 to x = 2 is rotated about the y-axis using the cylindrical shells method. This calculus problem involves integral calculus and substitution techniques. Suitable for undergraduate students.

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