In normalizing wave functions, the integration is over all space in which the wave function is defined.

a. Normalize the wave function x (a − x) y (b − y) over the range 0 ≤ x ≤ a, 0 ≤ y ≤ b. The element of area in two-dimensional Cartesian coordinates is dx dy; a and b are constants.

b. Normalize the wave function e^−2r / b sin θ sin∅ over the interval 0 ≤ r < ∞, 0 ≤ θ ≤ π, 0 ≤ ∅ ≤ 2π. The volume element in three-dimensional spherical coordinates is r² sinθ dr dθ d∅, and b is a constant.