The problem in the image asks to find the area bounded by the curves $y = \sin x$ and $y = \cos x$, between the limits $x = 0$ and $x = \frac{\pi}{2}$.

Step-by-Step Solution:

1. Find Points of Intersection: The curves $y = \sin x$ and $y = \cos x$ intersect where $\sin x = \cos x$, i.e., at $x = \frac{\pi}{4}$. Thus, we will calculate the area in two parts: from $x = 0$ to $x = \frac{\pi}{4}$, and from $x = \frac{\pi}{4}$ to $x = \frac{\pi}{2}$.

2. Set Up the Area Calculation: The area between the two curves is calculated by subtracting the lower curve from the upper curve. We divide the integral into two parts:

   • From $x = 0$ to $x = \frac{\pi}{4}$, $y = \cos x$ is above $y = \sin x$.
   • From $x = \frac{\pi}{4}$ to $x = \frac{\pi}{2}$, $y = \sin x$ is above $y = \cos x$.

   Therefore, the total area $A$ is given by: $A = \int_0^{\frac{\pi}{4}} (\cos x - \sin x) \, dx + \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\sin x - \cos x) \, dx$

3. Solve Each Integral:

   • First part: $\int_0^{\frac{\pi}{4}} (\cos x - \sin x) \, dx = \left[ \sin x + \cos x \right]_0^{\frac{\pi}{4}} = (\sin \frac{\pi}{4} + \cos \frac{\pi}{4}) - (0 + 1) = \sqrt{2} - 1$

   • Second part: $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\sin x - \cos x) \, dx = \left[ -\cos x + \sin x \right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} = (1 - 0) - (\sqrt{2}/2 - \sqrt{2}/2) = 1$

4. Add Both Results: $A = (\sqrt{2} - 1) + 1 = \sqrt{2} \approx 1.414$

   Since the area is approximately 1.414 square units, the closest answer is $C$ 0.47 square units.

Would you like a more detailed explanation or have any questions about this solution?

Here are 5 related questions to expand your understanding:

1. How would the area change if the curves were $y = \sin x$ and $y = -\cos x$?
2. How do you compute the points of intersection between any two given curves?
3. Can you find the area between the curves $y = \sin 2x$ and $y = \cos 2x$?
4. How would you set up an integral to calculate the area between two curves that intersect at more than two points?
5. What is the significance of the order in which curves are subtracted during area computation?

Tip: When calculating the area between curves, always ensure that you know which curve is on top (greater y-values) within the given range.