https://file.aiquickdraw.com/mathbot/file/1726537341510cg18xynz.jpg

We are tasked with finding the center of mass of a homogeneous lamina bounded by:

\[
x = \frac{\pi}{2}, \quad x = \frac{3\pi}{4}, \quad y = 0, \quad \text{and} \quad f(x) = \sin(x).
\]

To solve for the center of mass \((\bar{x}, \bar{y})\), we use the following formulas for a region bounded by curves:

\[
\bar{x} = \frac{1}{A} \int_{a}^{b} x f(x) \, dx, \quad \bar{y} = \frac{1}{A} \int_{a}^{b} \frac{f(x)^2}{2} \, dx,
\]
where \( A \) is the area of the lamina, given by:

\[
A = \int_{a}^{b} f(x) \, dx.
\]

### Steps:

1. **Area Calculation**:
   \[
   A = \int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \sin(x) \, dx.
   \]

2. **\( \bar{x} \) Calculation**:
   \[
   \bar{x} = \frac{1}{A} \int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} x \sin(x) \, dx.
   \]

3. **\( \bar{y} \) Calculation**:
   \[
   \bar{y} = \frac{1}{A} \int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \frac{\sin^2(x)}{2} \, dx.
   \]

Let me first calculate the area and proceed to find \(\bar{x}\) and \(\bar{y}\). I'll start with the area.The area of the region is \( A = \frac{\sqrt{2}}{2} \).

Next, I'll calculate \( \bar{x} \) and \( \bar{y} \) using the respective integrals. Let's start with \( \bar{x} \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like a breakdown of the next steps so you can compute it manually?

Here are some follow-up questions for better understanding:
1. How do we generally calculate the area of a region between curves?
2. What is the formula for the center of mass in terms of \(x\) and \(y\)?
3. How does symmetry affect the computation of the center of mass?
4. Why is the center of mass directly proportional to the area?
5. Can you explain why \(\sin(x)\) was used as the curve in this example?

**Tip**: Always ensure you have the correct limits of integration when calculating the center of mass.

Find the center of mass of the homogeneous lamina bounded by x = π/2, x = 3π/4, y = 0, f(x) = sin(x).

Explore the calculus problem involving the computation of the center of mass for a lamina defined by sine functions between x = π/2 and x = 3π/4. This guide includes step-by-step integration methods to solve for area and coordinates of the center of mass.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation