So imagine I have a curved roof (imagine one side of a sine wave) with an arbitrary length. Now imagine there is a rectangular carpet underneath with its center aligned with the peak of the roof. I need to find the volume, from the base of the rectangle up to the roof (so a cuboid with a curved top).
Thereafter, I need to change the orientation of the rectangle (with the center still aligned) by 25 degrees in the x-direction and see how that changes the total volume.
The length of the wave in x-direction would be 8.5m with a height of 0.03m. The wave can be described as a sine/cos curve. The rectangle dimension is 0.4mx0.5m.
How would I go about this? I added a quick sketch.
This is exactly what double integration solves. You have a $f(x,y) = A\sin\omega x + B$ that describes the height of the roof. The limits of integration will describe the area of the carpet.
$\int_{-\frac L2}^{\frac L2}\int_{-\frac W2}^{\frac W2} A\cos\omega x + B\ dx\ dy$
Changing the orientation of the carpet changes the limits of integration. Or, you can apply a change of variables that will re-orient your height function to your new variables.
$x = u\cos \phi + v\sin\phi\\ y = v\cos \phi - u\sin\phi$
($\phi$ is your rotation angle)
$\int_{-\frac L2}^{\frac L2}\int_{-\frac w2}^{\frac w2} A\cos(\omega u\cos\phi + \omega v\sin\phi) + B\ du\ dv$
If you don't like the change of variables, you can view the rotated region as a triangular region, a parallelogram, and a second triangle.