Rewrite the integral ∫ (0,1) ∫ (0,x) ∫(0, sqrt(1-x^2) z dzdydx in the order dydxdz.
is this gonna be ∫(0,1) ∫(0, sqrt(1-z^2) ∫(0,x) z dydxdz?
I'm inclined to say yes but to check, you should draw a change of variables diagram for each separate integral.
In terms of (x,y) since x is [0,1] and y is [0,x] then a change in variables w,ill give [0,x]
With regards to (x,z) the area of integration is the upper right quadrant of a unit circle and changing x and z only changes the x to a z through symmetry so this looks OK as well.
To be sure though, I would suggest you numerically integrate both expressions to see if you get the same answer. You could also use a computer package like Mathematica or Maple, but still integrating this expression is one way of getting evidence that the change is good.