Worked example - sifting and scaling. Evaluate I = integral(-inf, inf) x^2 cos(x) delta(2x - 4) dx. Step 1: find the zero: 2x - 4 = 0 at x = 2, with |g'| = 2, so delta(2x - 4) = delta(x - 2)/2. Step 2: sift: I = (1/2) x^2 cos(x) evaluated at x = 2 = (1/2) x 4 x cos(2) = 2 cos(2) ~ -0.832. Check by substitution: u = 2x - 4, dx = du/2, x = (u+4)/2; I = integral ((u+4)/2)^2 cos((u+4)/2) delta(u) du/2 = (1/2)(2)^2 cos(2) = 2 cos 2. Matches. Physics use: a hammer blow F(t) = J delta(t - t0) delivers impulse J instantaneously: integral F dt = J.