Chapter 5 | Integration
519
methods. The idea that the approximations of the area under the curve get better and better as n gets larger and larger is very important, and we now explore this idea in more detail. Values of n Approximate Area L n Approximate Area R n n =4 7.5 8.5
n =8
7.75
8.25
n =32
7.94
8.06
Table 5.1 Converging Values of Left- and Right-Endpoint Approximations as n Increases
Forming Riemann Sums So far we have been using rectangles to approximate the area under a curve. The heights of these rectangles have been determined by evaluating the function at either the right or left endpoints of the subinterval [ x i −1 , x i ]. In reality, there is no reason to restrict evaluation of the function to one of these two points only. We could evaluate the function at any point x i in the subinterval [ x i −1 , x i ], and use f ⎛ ⎝ x i * ⎞ ⎠ as the height of our rectangle. This gives us an estimate for the area of the form A ≈ ∑ i =1 n f ⎛ ⎝ x i * ⎞ ⎠ Δ x . A sum of this form is called a Riemann sum, named for the 19th-century mathematician Bernhard Riemann, who developed the idea. ⎤ ⎦ . Let Δ x be the width of each subinterval [ x i −1 , x i ] and for each i , let x i * be any point in [ x i −1 , x i ]. A Riemann sum is defined for f ( x ) as ∑ i =1 n f ⎛ ⎝ x i * ⎞ ⎠ Δ x . Definition Let f ( x ) be defined on a closed interval ⎡ ⎣ a , b ⎤ ⎦ and let P be a regular partition of ⎡ ⎣ a , b Recall that with the left- and right-endpoint approximations, the estimates seem to get better and better as n get larger and larger. The same thing happens with Riemann sums. Riemann sums give better approximations for larger values of n .We are now ready to define the area under a curve in terms of Riemann sums.
Definition
⎦ , and let ∑ i =1 n
f ⎛
⎞ ⎠ Δ x be a Riemann sum for
Let f ( x ) be a continuous, nonnegative function on an interval ⎡ ⎣ a , b ⎤
⎝ x i *
f ( x ). Then, the area under the curve y = f ( x ) on ⎡ ⎣ a , b ⎤
⎦ is given by
→∞ ∑ i =1 n
f ⎛
⎞ ⎠ Δ x .
A = lim n
⎝ x i *
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