Let f (x) = cos x and g(x) = sin x. Find the center of gravity (x, y) of the region between the graphs of f and g on 

0, 
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2

.

(x, y) = 

The center of gravity (centroid) \((x, y)\) of a region between two curves \(f(x)\) and \(g(x)\) over the interval \([a, b]\) is given by:

\[
x_{\text{centroid}} = \frac{1}{A} \int_a^b x [f(x) - g(x)] \, dx
\]
\[
y_{\text{centroid}} = \frac{1}{2A} \int_a^b [f(x)^2 - g(x)^2] \, dx
\]

where \(A\) is the area of the region, calculated as:

\[
A = \int_a^b [f(x) - g(x)] \, dx
\]

In this case:
- \(f(x) = \cos x\)
- \(g(x) = \sin x\)
- \(a = 0\) and \(b = \frac{\pi}{2}\)

### Step 1: Compute the area \(A\)
\[
A = \int_0^{\frac{\pi}{2}} (\cos x - \sin x) \, dx
\]

### Step 2: Compute \(x_{\text{centroid}}\)
\[
x_{\text{centroid}} = \frac{1}{A} \int_0^{\frac{\pi}{2}} x (\cos x - \sin x) \, dx
\]

### Step 3: Compute \(y_{\text{centroid}}\)
\[
y_{\text{centroid}} = \frac{1}{2A} \int_0^{\frac{\pi}{2}} (\cos^2 x - \sin^2 x) \, dx
\]

Let me compute these integrals for you.It appears there was an issue in calculating the integrals because the computed area \(A\) came out to be zero. This suggests a problem in the approach, likely due to a misunderstanding in how the functions interact in the given interval.

Let’s carefully revisit the integral and correct any possible mistake before retrying. Since \(f(x) = \cos(x)\) and \(g(x) = \sin(x)\), the functions intersect within the interval, meaning the region might need to be split.

I'll correct the setup and try again.It seems like I can't do more advanced data analysis right now. Please try again later. If you'd like, I can walk you through the process manually, or we can try other problems in the meantime!

Here are five related questions you could explore:
1. What are the steps to find the centroid for general curves?
2. How does the centroid change if \( f(x) \) and \( g(x) \) are shifted by constants?
3. What is the geometric meaning of the centroid for symmetric regions?
4. Can you compute the area between \( \cos(x) \) and \( \sin(x) \) on other intervals?
5. How does the centroid of a triangle differ from the centroid of a curve-bounded region?

**Tip:** When dealing with overlapping regions or crossing curves, always split the integral at the points of intersection.

Let f(x) = cos(x) and g(x) = sin(x). Find the center of gravity (x, y) of the region between the graphs of f and g on [0, π/2].

Explore how to compute the center of gravity (centroid) for the region between the graphs of f(x) = cos(x) and g(x) = sin(x) over the interval [0, π/2]. This problem involves key concepts in calculus, including integral calculus and centroid calculation.

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