Double Integral Lab

Iterated integrals: ∫∫ᴅ f(x,y) dA can be evaluated using either dx dy or dy dx, provided the chosen bounds describe exactly the same region D.

Rectangles: constant inner and outer limits. Triangles: inner limits can depend on the outer variable.

Polar coordinates: x = r cos θ, y = r sin θ, and dA = r dr dθ. The Jacobian factor r is essential.

Improper integrals: ∫₁^∞ x^(−p) dx converges for p > 1, with value 1/(p − 1); otherwise it diverges. The partial integral is (R^(1−p) − 1)/(1−p), for p ≠ 1, and ln R for p = 1.

Area has units²; a double integral of a height function over an area has units³.