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] 

To find the center of gravity (or centroid) \((x_c, y_c)\) of the region between the graphs of \( f(x) = \cos x \) and \( g(x) = \sin x \) on the interval \([0, \frac{\pi}{2}]\), we use the following formulas for the coordinates of the centroid:

\[
x_c = \frac{\int_a^b x \cdot (f(x) - g(x)) \, dx}{\int_a^b (f(x) - g(x)) \, dx}
\]
\[
y_c = \frac{\int_a^b \frac{f(x) + g(x)}{2} \cdot (f(x) - g(x)) \, dx}{\int_a^b (f(x) - g(x)) \, dx}
\]

### Step 1: Find the area of the region
The area of the region is given by:
\[
A = \int_0^{\frac{\pi}{2}} (\cos x - \sin x) \, dx
\]
Let's calculate this.

\[
\int_0^{\frac{\pi}{2}} (\cos x - \sin x) \, dx = \left[\sin x + \cos x\right]_0^{\frac{\pi}{2}}
\]
Evaluating this:
\[
A = \left( \sin\left(\frac{\pi}{2}\right) + \cos\left(\frac{\pi}{2}\right) \right) - \left( \sin(0) + \cos(0) \right)
= (1 + 0) - (0 + 1) = 0
\]
Thus, the total area is \( A = 0 \). Now we can proceed to calculating the coordinates of the centroid \( (x_c, y_c) \.



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].

Discover how to find the centroid (center of gravity) of the region between the curves of cos(x) and sin(x) on the interval [0, π/2]. This step-by-step guide covers calculus concepts such as definite integrals and the formulas for calculating centroids. Suitable for students at the undergraduate level.

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