512
Chapter 5 | Integration
Evaluate the sum indicated by the notation ∑ k =1 20
5.3
(2 k +1).
Approximating Area Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let f ( x ) be a continuous, nonnegative function defined on the closed interval ⎡ ⎣ a , b ⎤ ⎦ . We want to approximate the area A boundedby f ( x ) above, the x -axis below, the line x = a on the left, and the line x = b on the right ( Figure 5.2 ).
Figure 5.2 An area (shaded region) bounded by the curve f ( x ) at top, the x -axis at bottom, the line x = a to the left, and the line x = b at right.
How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval ⎡ ⎣ a , b ⎤ ⎦ into n subintervals of equal width, b − a n . We do this by selecting equally spaced points x 0 , x 1 , x 2 ,…, x n with x 0 = a , x n = b , and x i − x i −1 = b − a n for i =1, 2, 3,…, n . We denote the width of each subinterval with the notation Δ x , so Δ x = b − a n and x i = x 0 + i Δ x for i =1, 2, 3,…, n . This notion of dividing an interval ⎡ ⎣ a , b ⎤ ⎦ into subintervals by selecting points from within the interval is used quite often in approximating the area under a curve, so let’s define some relevant terminology. Definition A set of points P ={ x i } for i =0, 1, 2,…, n with a = x 0 < x 1 < x 2 <⋯< x n = b , which divides the interval ⎡ ⎣ a , b ⎤ ⎦ into subintervals of the form [ x 0 , x 1 ], [ x 1 , x 2 ],…, [ x n −1 , x n ] is called a partition of ⎡ ⎣ a , b ⎤ ⎦ . If the subintervals all have the same width, the set of points forms a regular partition of the interval ⎡ ⎣ a , b ⎤ ⎦ .
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