Labware - MA35 Multivariable Calculus - Two Variable Calculus

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Volume Under a Function Graph (Page: 1 | 2 )

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Just as in the Cartesian case, the definite integral of a function of two variables in polar coordinates represents the volume underneath its three-dimensional function graph.

Recall that the procedure for evaluating single variable integrals is to
1. approximate the area under the graph using rectangles; and
2. take the limit of the sum of the area of these rectangles as the number of rectangles approaches infinity.

Similarly, the volume underneath the function graph for a double integral can be found by
1. dividing the domain R into rectangles;
2. erecting rectangular prisms over these rectangles using the value of the function graph at the bottom-left vertex of each rectangle as the height;
3. taking the limit of the sum of the prismsí volumes as the number of rectangles in the domain aproaches infinity.

Demos