Evaluating ∬ (x²+y) dA over R=[0,2]×[1,3]

Calculus 3 · Axiom Academy

Set up a double integral over a rectangle as an iterated integral and evaluate it step by step. Evaluate the double integral over the rectangular region — that is, and . The region R is the axis-aligned rectangle . Nice work! You evaluated a double integral over a rectangle by turning it into two ordinary integrals, one after the other. Rectangle → constant limits: For every limit is a number, so . Hold the outer variable fixed: On the inner integral, integrate with respect to one variable and treat the other as a constant (here x^2 was constant while integrating in y ). Work inside-out: Finish the inner integral and substitute its limits before you touch the outer integral. Fubini's Theorem: Because f is continuous on a rectangle, the order may be swapped — integrating instead gives the same . This same inside-out routine handles any double integral over a rectangle — only the antiderivatives change.

This is the written version of the interactive lesson above. See the full Calculus 3 course.