Area between two curves as an integral
Suppose we wish to find the area of a region that is bounded above by the curve , below by the curve , and on the left and right by the lines and . The area can be approximated with a Riemann sum of rectangles, where the -th rectangle has height , where is the -th sample point in a partition of . Upon taking a limit of the Riemann sums (as the norm of the partition goes to zero), we obtain the value
.
In fact, this is how we define the area of the region between the curves, whenever throughout .
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