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1.1 Approximating areas  (Page 3/17)

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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 ] .

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 ( [link] ). We use the notation L n to denote that this is a left-endpoint approximation    of A using n subintervals.

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
A diagram showing the left-endpoint approximation of area under a curve. Under a parabola with vertex on the y axis and above the x axis, rectangles are drawn between a=x0 on the origin and b = xn. The rectangles have endpoints at a=x0, x1, x2…x(n-1), and b = xn, spaced equally. The height of each rectangle is determined by the value of the given function at the left endpoint of the rectangle.
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

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 ( [link] ).

A diagram showing the right-endpoint approximation of area under a curve. Under a parabola with vertex on the y axis and above the x axis, rectangles are drawn between a=x0 on the origin and b = xn. The rectangles have endpoints at a=x0, x1, x2…x(n-1), and b = xn, spaced equally. The height of each rectangle is determined by the value of the given function at the right endpoint of the rectangle.
In the right-endpoint approximation of area under a curve, the height of each rectangle is determined by the function value at the right of each subinterval. Note that the right-endpoint approximation differs from the left-endpoint approximation in [link] .

The graphs in [link] represent the curve f ( x ) = x 2 2 . In graph (a) we divide the region represented by the interval [ 0 , 3 ] into six subintervals, each of width 0.5. Thus, Δ x = 0.5 . We then form six rectangles by drawing vertical lines perpendicular to x i − 1 , the left endpoint of each subinterval. We determine the height of each rectangle by calculating f ( x i − 1 ) for i = 1 , 2 , 3 , 4 , 5 , 6 . The intervals are [ 0 , 0.5 ] , [ 0.5 , 1 ] , [ 1 , 1.5 ] , [ 1.5 , 2 ] , [ 2 , 2.5 ] , [ 2.5 , 3 ] . We find the area of each rectangle by multiplying the height by the width. Then, the sum of the rectangular areas approximates the area between f ( x ) and the x -axis. When the left endpoints are used to calculate height, we have a left-endpoint approximation. Thus,

A ≈ L 6 = ∑ i = 1 6 f ( x i − 1 ) Δ x = f ( x 0 ) Δ x + f ( x 1 ) Δ x + f ( x 2 ) Δ x + f ( x 3 ) Δ x + f ( x 4 ) Δ x + f ( x 5 ) Δ x = f ( 0 ) 0.5 + f ( 0.5 ) 0.5 + f ( 1 ) 0.5 + f ( 1.5 ) 0.5 + f ( 2 ) 0.5 + f ( 2.5 ) 0.5 = ( 0 ) 0.5 + ( 0.125 ) 0.5 + ( 0.5 ) 0.5 + ( 1.125 ) 0.5 + ( 2 ) 0.5 + ( 3.125 ) 0.5 = 0 + 0.0625 + 0.25 + 0.5625 + 1 + 1.5625 = 3.4375.
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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