Volume Interpretation

    If on the interval the single integral can be interpreted as the area under the curve over The double integral



has a similar interpretation in terms of volume; since it is natural to define the total volume under the surface as the limit of Riemann sums as the norm tends to 0; of course provided, on the rectangular region

Proposition (Volume Interpretation of the Double Integral) If on the rectangular region then the product is the volume of a parallelepiped (a box) with height and base area The Riemann sum  



provides an estimate of the total volume under the surface over and if is continuous, we expect the approximation to improve by using more refined partitions. That is, the volume under over the domain is given by



when on the rectangular region

Example (Volume Interpretation) Find the volume of the solid region bounded below by the given rectangle in the -plane and above by the graph of the given surface.

(a) Find the volume of the solid bounded below by the rectangle in the and above by the graph of

    Solution. We compute  










(b) Find the volume of the solid bounded below by the rectangle in the and above by the graph of

    Solution. We compute  










Note in this example we used the formulas and

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