Defining Double Integration with Riemann Sums | Volume under a Surface

We generalize the ideas of integration from single-variable calculus to define double integrals. The big idea in single variable calculus was to chop up the region into a sum of little rectangles called the Riemann sum which was an approximation for the area under a function. Then we took a limit of the Riemann sum to define the definite integral. We do much the same here, looking to find a formula for the volume under a surface. Now a rectangular region in the domain is broken up into a lot of little prisms and the sum of those volumes is the Riemann sum. Take the limit as the sizes in that partition goes to zero and this defines a double integral.

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