Chapter 5 | Integration
513
We can use this regular partition as the basis of a method for estimating the area under the curve. We next examine two methods: the left-endpoint approximation and the right-endpoint approximation.
Rule: Left-Endpoint Approximation On each subinterval [ x i −1 , x i ] (for i =1, 2, 3,…, n ), construct a rectangle with width Δ x and height equal to f ( x i −1 ), which is the function value at the left endpoint of the subinterval. Then the area of this rectangle is f ( x i −1 )Δ x . Adding the areas of all these rectangles, we get an approximate value for A ( Figure 5.3 ). We use the notation L n to denote that this is a left-endpoint approximation of A using n subintervals. (5.6) A ≈ L n = f ( x 0 )Δ x + f ( x 1 )Δ x +⋯+ f ( x n −1 )Δ x
= ∑ i =1 n
f ( x i −1 )Δ x
Figure 5.3 In the left-endpoint approximation of area under a curve, the height of each rectangle is determined by the function value at the left of each subinterval.
The second method for approximating area under a curve is the right-endpoint approximation. It is almost the same as the left-endpoint approximation, but now the heights of the rectangles are determined by the function values at the right of each subinterval. Rule: Right-Endpoint Approximation Construct a rectangle on each subinterval [ x i −1 , x i ], only this time the height of the rectangle is determined by the function value f ( x i ) at the right endpoint of the subinterval. Then, the area of each rectangle is f ( x i )Δ x and the approximation for A is given by (5.7) A ≈ R n = f ( x 1 )Δ x + f ( x 2 )Δ x +⋯+ f ( x n )Δ x
= ∑ i =1 n
f ( x i )Δ x .
The notation R n indicates this is a right-endpoint approximation for A ( Figure 5.4 ).
Made with FlippingBook - professional solution for displaying marketing and sales documents online