Does the average height change after we squished the base?

Question

My question is, for an arbitrary surface $z = f(x,y)$, if the base coordinate x and y has experienced some linear transformation. Can I multiply the old calculated average height in Z direction to the new base area in XY to get the new volume?

Answer 1

Yes if you update $f(x,y)$ so it gives the same height over the transformed point as the old $f(x,y)$ did. The average height is $$\overline h=\frac {\int f(x,y)dxdy}{\int dxdy}$$ where $\int dxdy$ is the area of the base. If you apply a linear transformation to get to $(u,v)$ the area of the base will be $\int dudv$, which will be the old area times the Jacobian. Each small slice of the volume will have its area multiplied by the same amount.