Math basics › Multivariable calculus
Double and triple integrals
Inside out
Do the inner integral first, treating the other variable as a constant. Then integrate what is left.
The bounds are the hard part
Sketch the region. The inner limits can depend on the outer variable; the outer limits must be plain numbers.
Switching the order
Redraw the region and read the bounds the other way. It can turn an impossible inner integral into an easy one.
Changing coordinates
Polar: dA = r dr dθ. Cylindrical: dV = r dz dr dθ. Spherical: dV = ρ² sin φ dρ dφ dθ. The extra factor is what people forget.