How would I find the surface area in red using integrals? What operation do I have to do? Especially for the surface area above x-axis, since I know the the answer for the one under the x-axis is "1".

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Area between two curves f(x) and g(x) is the integral of $f(x) - g(x)$ where f(x) is the upper one and g(x) is the lower one.

Since they swap places, i.e. cos(x) is the top one in one region and the bottom one in another, you have to integrate in two parts, one where cos(x) is on top and one where sin(x) is on top.
interestingly you can break it up into two intervals for when sinx is larger and when cosx is larger. Namely $0,π/4$ and (π/4,π\]. Sinx - cosx for first interval and cosx - sinx for second.

Or you can integrate |sinx - cosx|
You have to break it into 2 parts and integrate

From 0 to π/4 Cosx is above Sinx thus, integrate cosx-sinx from 0 to π/4

From π/4 to π Sinx is above cosx thus the integral will be sinx-cosx from π/4 to π

The desired area is the summation of these 2 integrals
I would use a rieman sum to calculate the area under the curve... You would do this by using trapezoids with left or right stepping...

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