This lecture develops the Fundamental Theorem for iterated double integrals in multivariable calculus, establishing that a double integral over a plane region R (defined as the limit of a double Riemann sum) can be evaluated as an iterated antiderivative process, provided the integrand f(x,y) is continuous on R. The theorem is presented as a direct analogue of the single-variable Fundamental Theorem of Calculus: just as a definite integral (a limit of sums) can be computed via an antiderivative, a double sum over a region bounded above and below by curves y = G1(x) and y = G2(x) (with x ranging from a to b) can be computed by holding one variable fixed, integrating with respect to the other between region-dependent limits, and then integrating the resulting single-variable function over the outer bounds. This connects two conceptually distinct constructions — the double integral as a limit of sums (from prior lecture) and the iterated antiderivative (double integral as repeated single integration) — showing they yield the same numerical result whenever the limit exists.
This lecture develops the Fundamental Theorem for iterated double integrals in multivariable calculus, establishing that a double integral over a plane region R (defined as the limit of a double Riem…