one question about the substitution in multiple integrals

[bingliantech]

I konw the Jacobian of the coordinate transformation measures how much the transformation is expanding or contracting the area around a point in G as G is transformed into R.My question is why we need the Jacobian in the transformation,the deltaA which represent small area go into almost zero in the integral,and the deltaS which represnt the transformed small area also go into zero,why we need the Jacobian to measure the multiple.