Step 1:

Based on the graphs of [latex]\sin(x)[/latex] and [latex]\cos(x)[/latex] we can see that between the interval of [latex]x = 0[/latex] and [latex]x = \frac{\pi}{4}[/latex], the cosine function is the upper function, and the sine function is the lower function.

Thus, to get the area between these curves, we will solve the following definite integral:

[latex]\large{\int_{0}^{\frac{\pi}{4}}\cos(x) – \sin(x) dx}[/latex]

Using our knowledge of common integrals, we have:

[latex]\large{ = (\sin(x) + \cos(x))|_{0}^{\frac{\pi}{4}}}[/latex]

Step 2:

Now we can apply the fundamental theorem of calculus to evaluate this definite integral. This states that:

[latex]\large{\int_{a}^{b}f(x) dx = F(b) – F(a)}[/latex]

Applying the fundamental theorem of calculus, we have:

[latex]\large{\int_{0}^{\frac{\pi}{4}}\cos(x) – \sin(x) dx}[/latex]

[latex]\small{ = [\sin(\frac{\pi}{4}) + \cos(\frac{\pi}{4})] – [\sin(0) + \cos(0)]}[/latex]

[latex]\large{ \approx 1.414 – 1}[/latex]

[latex]\large{\approx 0.414}[/latex]

Thus, the area between the functions [latex]y = \sin(x)[/latex] and [latex]y = \cos(x)[/latex]

between the specified intervals is approximately 0.414 units.

Solution Completed!