To find the volume of the solid obtained by rotating the region bounded by the curve $y = \sqrt{x}$, the x-axis, and the line $x = 4$ about the y-axis, we use the method of cylindrical shells.

Step 1: Express $x$ in terms of $y$

Given the curve $y = \sqrt{x}$, we first solve for $x$ in terms of $y$: $y = \sqrt{x} \implies x = y^2$

Step 2: Set up the volume integral using the shell method

The formula for the volume of a solid of revolution using the method of cylindrical shells is: $V = 2\pi \int_{a}^{b} ( \text{radius} ) \times (\text{height}) \, dy$

• Radius: The radius of a cylindrical shell is the distance from the axis of rotation (the y-axis), which is simply $x = y^2$.
• Height: The height of the cylindrical shell is given by the function $x = y^2$, ranging from $y = 0$ to $y = 2$ (since $y = \sqrt{x}$ and $x = 4$).

The volume integral becomes: $V = 2\pi \int_{0}^{2} y \times y^2 \, dy$ $V = 2\pi \int_{0}^{2} y^3 \, dy$

Step 3: Evaluate the integral

To evaluate the integral, first compute the antiderivative of $y^3$: $\int y^3 \, dy = \frac{y^4}{4}$

Now, apply the limits of integration from 0 to 2: $V = 2\pi \left[ \frac{y^4}{4} \right]_{0}^{2}$ $V = 2\pi \left( \frac{2^4}{4} - \frac{0^4}{4} \right)$ $V = 2\pi \left( \frac{16}{4} \right) = 2\pi \times 4 = 8\pi$

Final Answer

The volume of the solid is $\boxed{8\pi}$ cubic units.

Would you like more details or have any questions?

Here are some related questions for further practice:

1. How would the volume change if the region were rotated about the x-axis instead?
2. What is the volume if the region is rotated about the line $x = 6$ instead of the y-axis?
3. How would you find the surface area of the solid formed by rotating the same region about the y-axis?
4. What if the curve was $y = x^2$ instead of $y = \sqrt{x}$? How would the volume change?
5. How does the volume calculation differ if the boundary is extended to $x = 9$?

Tip: The shell method is particularly useful when the solid of revolution involves a function that is easier to express in terms of the radius and height relative to the axis of rotation.