HigherIntegration - Higher Dimensional Integration Jed...

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Higher Dimensional Integration Jed Keesling So far we have only considered integration of a function on the real line. It is easy enough to generalize this to higher dimensions. However, you need to be alerted to some caveats. This will be a topic that will be covered more thoroughly in MAC 2313 or MAC 3474. Consider the function ! ! , ! = ! ! + ! ! . We would like to determine the volume that is under the graph of this function above the xy –plane over the square 3 , 3 × [ 3 , 3 ] . The graph is given below We break up the sides of the square into small intervals and this breaks up the square into a grid of rectangles. We approximate the volume over each of these rectangles by the formula ! = ! ( ! ! , ! ! ) ! ! where ! and ! are the sides of the small rectangle and the point ( ! ! , ! ! ) is a point in this rectangle. We express the limit of the sums of these small volumes as an integral. In the above example it would be given in the following form.
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