The area under the curve y=|cosx−sinx|, 0≤x≤π2, and above x-axis is :

To find the area under the curve y=|cosx−sinx|, 0≤x≤π/2, and above the x-axis, we need to integrate the function from 0 to π/2. However, because the function is an absolute value, we need to consider when the function inside the absolute value is positive and when it is negative.

1. First, we need to find the x-value where cosx = sinx in the interval [0, π/2]. This occurs when x = π/4.

2. Now we split the integral into two parts: one from 0 to π/4 and the other from π/4 to π/2.

3. From 0 to π/4, cosx > sinx, so |cosx - sinx| = cosx - sinx. We integrate this from 0 to π/4.

4. From π/4 to π/2, sinx > cosx, so |cosx - sinx| = sinx - cosx. We integrate this from π/4 to π/2.

5. The total area under the curve is the sum of these two integrals.

Let's calculate:

∫ from 0 to π/4 (cosx - sinx) dx + ∫ from π/4 to π/2 (sinx - cosx) dx

= [sinx + cosx] from 0 to π/4 + [-cosx - sinx] from π/4 to π/2

= [(1/√2 + 1/√2) - (1 + 0)] + [-(0 + 1/√2) - (-1/√2 - 1/√2)]

= √2 - 1 - 1/√2 + √2

= 2√2 - 1 - 1/√2

So, the area under the curve y=|cosx−sinx|, 0≤x≤π/2, and above the x-axis is 2√2 - 1 - 1/√2.