https://file.aiquickdraw.com/mathbot/file/17290545648317phozuiu.jpg

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.

Find the volume of the solid with the given base and cross-sections by first finding the formula for the area of a typical cross-section and then integrating over the appropriate region and variable. The base of the region is enclosed by y = 5 - x^2 and y = 2, and the cross-sections perpendicular to the y-axis are squares.

Learn how to find the volume of a solid with a base enclosed by y = 5 - x^2 and y = 2, where the cross-sections perpendicular to the y-axis are squares. This problem uses integral calculus to compute the volume by integrating the area of the cross-sections.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation