We are asked to find the volume of a solid with a given base and square cross-sections. Let's go through the solution process step by step.

Problem Summary:

We are given:

• The base of the region is enclosed by $y = 5 - x^2$ and $y = 2$.
• The cross-sections perpendicular to the $y$-axis are squares.

We need to:

1. Find the formula for the area of a typical cross-section (which is a square).
2. Set up and evaluate the integral to find the volume.

Step 1: Identify the bounds for $x$

The cross-sections are perpendicular to the $y$-axis, so we express $x$ in terms of $y$. For the base, the curve $y = 5 - x^2$ can be rewritten as: $x^2 = 5 - y \quad \text{or} \quad x = \pm \sqrt{5 - y}$ Thus, the width of the square cross-section at a given $y$ will be $2\sqrt{5 - y}$.

Step 2: Find the area of a typical cross-section

Since the cross-section is a square, the side length of the square at a given $y$ is $2\sqrt{5 - y}$. The area of the square is: $\text{Area} = \left( 2\sqrt{5 - y} \right)^2 = 4(5 - y) = 20 - 4y$

Step 3: Set up the integral

The volume of the solid can be found by integrating the area of the cross-sections along the $y$-axis from $y = 2$ to $y = 5$, as these are the bounds of the base. Therefore, the volume $V$ is: $V = \int_{2}^{5} (20 - 4y) \, dy$

Step 4: Evaluate the integral

Now we compute the integral: $V = \int_{2}^{5} (20 - 4y) \, dy = \left[ 20y - 2y^2 \right]_{2}^{5}$ Evaluating this: $V = \left( 20(5) - 2(5)^2 \right) - \left( 20(2) - 2(2)^2 \right)$ $V = (100 - 50) - (40 - 8) = 50 - 32 = 18$

Final Answer:

The volume of the solid is $18$ cubic units.

Would you like more details or have any questions about this process?

Here are 5 related questions to deepen your understanding:

1. How would the volume change if the cross-sections were rectangles instead of squares?
2. What if the cross-sections were semicircles instead of squares?
3. How would the integral change if the base was enclosed by different curves, such as $y = 3 - x^2$?
4. How can you find the volume if the cross-sections were not perpendicular to the $y$-axis but to the $x$-axis?
5. What methods could you use to approximate the volume if the function was more complex?

Tip: When dealing with solids of revolution or other shapes, always start by analyzing the shape of the cross-section and relate it to the variable of integration.