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What is an integral?


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August 29, 2011 5:41PM

The integral of a function between a and b can be thought of as the area of the region bounded by the graph of the function and the x-axis between x=a and x=b. When f(x) is negative, the integral is negative.

The integral can be defined many ways, the most straightforward of which is summarized as follows:

Partition the interval [a,b] into n subintervals. On each interval, find the the least upper bound of the of the function. Then add up the areas of the rectangles obtained by multiplying the the width of each interval by the supremum on that interval. This is called the Upper Sum. The Lower Sum is found similiarly, except the supremum is replaced by the infimum. If the least upper bound of the set of lower all lower sums is equal to the greatest lower bound of the set of all upper sums, then the function is integrable on the interval, and the common number is called the integral.