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Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Chapter 5: The IntegralSection 5.1: Approximating and Computing Area

Jon Rogawski

Calculus, ETFirst Edition




Example, Page 3082. Figure 14 shows the velocity of an object over a 3-min interval. Determine the distance traveled over the intervals [0, 3] and [1, 2.5]. Remember to convert mph to mi/min.



Combining Figures 2 and 3, we may construct N intervals of equal width, Δx = (b – a)/N, over the interval [a, b] such that the right endpoints of the intervals are given by:


Since we used rectangles of common width, the approximate areaunder the curve becomes:

Using the second rule of linearity of summations, we obtain

Some texts refer to this method as the Rectangular ApproximationMethod or RAM.

Example, Page 3086. Use the following table of values to estimate the area under the graph of f (x) over [0, 1] by computing the average of R5 and L5.



So far, we have considered only approximations using the value of f (x)on the right of the rectangle, but we may also use the value at the left or the midpoint, as shown below.


The summations we use for the left-hand and midpoint approximationsare respectively:

The Rectangular Approximation Methods may further be abbreviatedas RRAM, LRAM, and MRAM to indicate using the y-values on the right, left, and midpoint respectively.


Figures 7 and 8 illustrate using midpoint, left and right approximationsof the area under a parabola and also, the amount of error in using eachmethod.


In Figures 8 and 9, we observe using RRAM and LRAM to approximate the area under curves that are either continuously increasing or decreasing. Such curves are said to be monotonic.Notice RRAM underestimatesareas for functions thatare monotonic decreasing and over-estimates those mono-tonic increasing. LRAMdoes the opposite.

Example, Page 30810. Estimate R2, M3, and L6 for the graph in Figure 16.



The benefit of using more rectangles illustrated in Figure 10 leads toTheorem 1:

Example, Page 30810. Estimate R2, M3, and L6 for the graph in Figure 16.


Homework

Homework Assignment #34 Read Section 5.2 Page 308, Exercises: 1 – 25(EOO)
