Lecture The Definite Integral (1 of 2) MTH 124

Transcription

1 We now begin our study of the integral or anti-derivative. Up to this point we have explored how to calculate the rate of change of a quantity. We now consider a related problem. That is, given the rate of change of a function, what can we know about the function? Let us start with a familiar example. Suppose we decide to head south for spring break. After traveling for 4 hours at 60 miles per hour we pull over at a rest stop. Below we give a simplified plot of our velocity. 1. How far have we traveled in these 4 hours? 2. Give a graphical representation of this distance in plot above. Okay, but what about if we change our velocity? For example suppose now we hit some traffic and our velocity follows the plot below. 3. Determine the distance we ve traveled over the first 6 hours and illustrate the distance graphically on the figure above. 95

2 What we ve done so far is fine and dandy, but as you may have noticed this isn t very realistic. When we travel in a vehicle a plot of our velocity would not look like the plots above. For example we don t instantaneously change velocity from 60 mi/hr to 40 mi/hr (unless of course you re able to teleport...) as is seen in the second plot. Instead, a more realistic picture of our velocity might look something like the figure below. Given this figure it is not immediately obvious that we can use the techniques we previously used to determine the distance we ve covered in the first 6 hours. However, the ideas we used above can help us here. That is, what we took advantage of in the previous problems was the idea that the area under the curve represents the change in distance over the given time interval. Due to the non-linearity of the curve we can no longer determine the exact area, however using rectangles we can determine an approximation of this area. 4. Use the areas of rectangles to estimate the distance traveled over 6 hours. Sketch your rectangle(s) in the figure above. Is your estimate an over-estimate, under-estimate, or undecidable? Justify why this approach makes sense. 96

3 The Riemann Sum We could use many different geometric shapes to approximate the area under the curve above. Fortunately for us a cohesive and well-defined system of approximating the area under a curve has already been developed. Moreover, using this method we can segue from a method that approximates the area under a curve over an interval to a method that gives the exact area under the curve. A Riemann sum approximates the area under a curve using the area of rectangles of equal width whose heights are determined by the function values. When we use left endpoint of each subinterval for the rectangle height our area approximation is known as a left Riemann sum (LHS). Similarly when we use right endpoint of each subinterval for the rectangle height our area approximation is known as a right Riemann sum (RHS). The figure below gives an example of a LHS (left) and RHS (right) over the interval [0, 10]. Mathematically the sum of these rectangles is described below. Riemann Sum If f is a continuous function, the left Riemann sum (LHS) and right Riemann sum (RHS) with n equal subdivisions of f over the interval [a, b] are defined to be and n 1 LHS = f(x k ) x = f(x 0 ) x + f(x 1 ) x f(x n 1 ) x, RHS = k=0 n f(x k ) x = f(x 1 ) x + f(x 2 ) x f(x n ) x, k=1 where a = x 0 < x 1 <... < x n = b and x = b a n. The definition above gives us instructions to determine the LHS or RHS of a function given an interval and number of rectangles desired. Put another way, to determine a LHS or RHS, we need to know the function, f(x) number of rectangles, n, and interval we re summing the rectangles over, [a, b]. Given this information we can determine our rectangle width, x, using n and [a, b]. 97

4 Example 1 Given the figure below determine the LHS of g(x) over [1, 3] with n = 4. 98

5 5. Given the figure below determine the RHS of f(x) over [0, 4] with n = Determine the LHS of g(x) = x 2 over [ 2, 2] with n = Determine the RHS of h(x) = x over [0, 1] with n = 5. 99

6 Given the figures below sketch and calculate the specified Riemann sum. 8. (a) LHS over [0, 8] with n = 4 (b) LHS over [ 4, 2] with n = 6 (c) RHS over [0, 4] with n = 2 (d) RHS over [1, 4] with n = 6 100

7 Exact Area Using Geometric Formulas As we ve now seen Riemann sums allow us to approximate the area under a curve which is particularly useful for non-linear functions. However, if we are given linear or piecewise linear functions we can use basic area formulas to determine the exact area. Example 2 Given the figure below, determine the area under the curve over [1, 5] Given the figure below, determine the area under the curve over [ 2, 5]. 1 The phrase area under the curve refers to the area between the function and the horizontal axis. 101