Approximating ∬ eˣ²⁺ʸ² dA over Unit Disk

Calculus 3 · Axiom Academy

EXAMPLE Approximating Over the Unit Disk No closed-form antiderivative — expand e^ xy as a Taylor series and integrate term by term. Estimate the double integral , where D is the closed unit disk . The integrand e^ xy has no elementary antiderivative in x or y , so we cannot integrate it in closed form. Instead, replace e^ xy with its Maclaurin series and integrate each power over the disk. The region D : the unit disk , area . Nice work. When an integrand has no closed-form antiderivative, a Taylor series turns one hard integral into a sum of easy polynomial integrals. Series beats "no antiderivative": , and each is an elementary polynomial moment. Symmetry kills the odd terms: D is symmetric under , so for every odd n — half the work disappears for free. Even moments over the disk: and , giving the leading corrections and . Result: , matching a direct numerical integral ( ) to within . Adding the tiny degree-4 term pushes the estimate to 3.2077 — the series converges fast because on the disk.

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