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We will develop the definite integral as a means to calculate the area of certain regions in the plane. Given two real numbers a < b and a function f (x) defined on the interval [a, b] , define the region R(f, a, b) to be the set of points (x, y) in the plane with a≤x≤b and with y between 0 and f (x) . Note that this region may lie above the x -axis, or below, or both, depending on whether f (x) is positive or negative. In computing the area of R(f, a, b) , it will be convenient to count the regions above the x -axis as having "positive area", and those below as having "negative area".
We can split up the interval [a, b] into n smaller intervals (for some integer n ) of width Δx = (b - a)/n . Let
s _{i} = a + i(Δx) |
for i = 0, 1,…, n , so that the n intervals are given by [s _{0}, s _{1}],…,[s _{n-1}, s _{n}] .
Let M _{i} be the maximum value of f (x) on the interval [s _{i-1}, s _{i}] . Similarly, let m _{i} be the minimum value of f (x) on the interval [s _{i-1}, s _{i}] . Consider the region made up of n rectangles, where the i -th rectangle is bounded horizontally by s _{i-1} and s _{i} and vertically by 0 and M _{i} . As shown below, this region contains R(f, a, b) .
Moreover, we know how to compute the area of this region. It is simply
(M _{1}) + (M _{2}) + ^{ ... } + (M _{n}) = M _{i} |
We denote this n th upper Riemann sum by U _{n}(f, a, b) . Replacing M _{i} in the above with m _{i} , we obtain a region contained in R(f, a, b) .