ESTIMATING AND CALCULATING AREAS OF REGIONS BETWEEN THE X-AXIS AND THE GRAPH OF A CONTINUOUS FUNCTION v.07

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1 ESTIMATING AND CALCULATING AREAS OF REGIONS BETWEEN THE X-AXIS AND THE GRAPH OF A CONTINUOUS FUNCTION v.7 In this activity, you will explore techniques to estimate the "area" etween a continuous function and the x- axis on a closed interval. To understand these ideas, we start considering the area of the region etween a straight line, represented y the function y f x and the x-axis on the interval 1, 5. Its graph is shown elow. 1, y f x 5, Notice that the area of the region in consideration, due to its shape, can e calculated exactly. The area is. For general regions, it is impossile to reak them up into rectangles and triangles to get an exact area. However, most of the time the area of a region can e approximated using triangles or rectangles. You will explore different ways to estimate this area. Below you find a rief explanation of each technique. All the techniques rely on sudividing the domain in consideration, [1,5], into equal suintervals. It is important to make clear that the only reason why equal suintervals are used is to make the computational part easier. The examples elow deal with four sudivision of the interval [1,5] Before any calculations are done, let's find some ounds for the area (underestimate, overestimate). These are values etween which the actual area is. There are different ways to calculate under and over estimates. The simplest one consists on taking the smallest rectangle with area greater than or equal to the area of the region, and the largest rectangle with area less than or equal to the area of the region. Notice that this has to do with the largest and smallest "height" of the function on the given closed interval. (5,) (1,)

2 Integration Estimating Areas v7.n The area in consideration is etween and. This is written as Area _ GRID (Boxes) Using the scale in the coordinate axes make a grid. Estimate the numer of rectangles needed to cover the region and then estimate the total area y using the area of a rectangle and the numer of rectangles estimated to fill up the region. y f x Area of each rectangle Approximate numer of rectangles to fill up the region Approximate area of the region For the next three techniques the heights of the function at each of the sudivision points are needed. Rememer that to facilitate our calculations the end points of the sudivisions are at equal distance called x ("delta x"). This value depends upon the length of the interval and the numer of sudivisions. In our case the length of the interval is and we want four sudivisions. Hence, x 1 (length of the interval divided y the numer of sudivisions). f 5 f f f f 1 x RECTANGLES Divide the region into rectangles using the left (right) hand points of each sudivision or suinterval. At each end point of the suintervals find the height and construct a rectangle toward the right direction of width x. Now add their areas up. The result gives an approximation of the total area. The graph elow shows the rectangles using the left hand points. Since the rectangles are constructed using the left-hand points of the suintervals, the sum is called the Left-Hand (Riemann) Sum. In a similar way the Right-Hand Sums can e calculated.

3 Integration Estimating Areas v7.n f 5 f f f f 1 x Area Left hand sum = f 1 x f x f x f x = *1+*1+*1+5*1 =1 Summation Notation Sums with pattern of formation can e written in a simplified manner using the summation notation (sigma notation). For instance, f 1 x f x f x f x i1 f i x This notation will e very useful in calculating sums when there is a large numer of terms. EXERCISE 1 Approximate the area using the right-hand sums. TRAPEZOID Rememer that the area of a trapezoid with heights A, B, and width C is given y A B C. Verify this formula. A A A C B To estimate the area of the region using trapezoids, use the heights of the function to make the trapezoids and then add the areas up. f 5 f 1 f f f x

4 Integration Estimating Areas v7.n Area Sum of the areas of the trapezoid f 1 f x x f f x f f f 1 f f f f 5 Why? x f f 5 x Using summation notation Area INDIVISIBLES x f 1 f 5 i f i For our discussion, an indivisile (it can not e divided any more) is simply a line segment. The area of the region in consideration is estimated y constructing a rectangle of width the same as the length of the interval [1,5], and height equal to the average of the "heights" of the indivisiles determined y the end points of the sudivisions. The height of each indivisile is the value of the function at each of the end points of the sudivisions. f 5 f f f f 1 x Average height is f 1 f f f f 5 5 y f x The area of the region is approximately the area of the shaded rectangle. The area of the rectangle otained aove is the Average value of the heights multiplied y the length of the interval. These oservations lead to the following powerful oservation The estimates of the area etween the function and the x-axis, using any of the methods descried aove, ecome ecome more accurate as the numer of sudivisions of the interval increase. THE SYMBOL a f x x AND AREA The value of the area etween the graph of the continuous function y f x and the x-axis on the interval a, is denoted a f x x

5 Integration Estimating Areas v7.n AVERAGE VALUE OF A CONTINUOUS FUNCTION ON AN INTERVAL [a,] One of the techniques used to estimate the area was indivisiles. It was oserved that the average value of the indivisiles (heights of the function) multiplied y the length of the interval is approximately the area under the curve. The actual area corresponds to the average of "all" the heights of the function on the interval multiplied y the length of the interval. This oservation leads to the following expression to calculate the average value of a continuous function on a closed interval. EXERCISE Average height of a continuous function y f x on a, a f x x a Consider the region etween the x-axis and the graph of the function g x x 1 on the interval [,1], whose graph is displayed elow. i) Find an overestimate and underestimate for the area using rectangles. ii) Estimate the area using seven sudivisions a. Grid technique.. Left-hand rectangles. c. Indivisiles. d. trapezoids. For questions (ii) (a-d), also write the expression using summation notation. Then use the calculator to find those sums. y g x iii) Use the graphing calculator and the graph of the function to find the actual value of the area of the region. NOTE: This area can e calculated using your graphing calculator. This is done y entering the function as Y=, graph it setting a window where you can see the region in consideration. After the region is displayed in your calculator, hit F5:Math, 7: f x x. Now the calculator asks for lower limit and upper limit. Those are the values of the interval that is the domain of the function, in this case and 1. Now you will see the region shaded and the value of the area at the ottom.

6 Integration Estimating Areas v7.n EXERCISE (Areas with Negative heights). Consider the region etween the x-axis and the graph of the function h x x x on the interval 1, y h x Oserve that in this case the heights are negative values and consequently the "area", the way it is calculated, is a negative numer. Don't panic, it is okay. It makes sense since these numers are otained y multiplying width and height. i) Find and overestimate and lower estimate for the area. ii) Estimate the area using rectangles and twelve sudivisions (use summation notation and the calculator). iii) Estimate the area using indivisiles and twelve sudivisions (use summation notation and the calculator). iv) Calculate the actual area using the graph in the calculator. v) Use the answer from part (iv) to calculate the average value of the function y= h(x) on the interval 1, 5 Notice that the standard idea of area of a region corresponds to the asolute value of the value(s) found aove. This in turn corresponds to the area of the asolute value of the function. This discussion leads to the definition of the Asolute Area of a region ounded y a continuous function y=f(x) and the x-axis on an interval [a,]. The asolute area is the area of the region enclosed y y= f(x) and the x-axis on [a,]. EXERCISE (Area with negative and positive heights) Consider the region elow that is ounded y the x-axis and the graph of y sin x on the interval, 5 Π.

7 Integration Estimating Areas v7.n i) Find an overestimate and underestimate for the area of the region. ii) Estimate the area using five sudivisions and any two of the techniques discussed aove. iii) Use the calculator to find the area of the region. iv) Calculate the asolute area of the region. v) Estimate the average value of the function on the given interval. EXERCISE 5 Consider the region etween the x axis and the graph of the function y x x x on the interval, 5. Use the calculator to a. Find the area of the region. Find the asolute area of the region c. Is there a value of so that y x x? If your answer is yes, estimate it d. Is there a value of c so that cy x x 15? If you answer is yes, estimate it. BASIC FORMULAS TO CALCULATE INTEGRALS All the functions y f x considered here are continuous on the given interval (no reaks, no jumps). These formulas are used to calculate the integrals algeraically. First of all some notation. Each of expressions elow are of the form a f x x F x a. They represent the integral of the function y f x on the interval [a,]. The right hand side of that expression F x a F()-F(a). 1. a C x Cx a, C is a constant. a x n x xn 1 n 1 a, n 1

8 Integration Estimating Areas v7.n. a 1 x x Ln x a, a, positive real numers. a m x x mx Ln m a, m a positive constant 5. a sin x x cos x a. a cos x x sin x a EXAMPLE 1 1. x x x x x.5x 1 Ln Ln Ln x x 1 x 1 x x x1 1 1.

9 Integration Estimating Areas v7.n a. x x iii) Use the calculator to find the value of the integral. Π. Π sin x x c. 5 1 x x d. x x Rules to operate with integrals I. c d a f x x a c d f x x, where a is a constant II. c d f x ± g x x= c d f x x ± c d g x x III. a f x x c a f x x a c f x x EXAMPLE a. 1 x x 1 x x Rule I x 1 Basic formula *. x x x x x x x Rule II x x x x Rule I x x 1 1 Basic formula c. x x x x x x x Rule II x x x 1 x x Ln x EXERCISE 7

10 1 Integration Estimating Areas v7.n Consider the function f x x 1 on the interval 1,. The area of the region etween the x-axis and the function is shown elow f x x 1.1. Determine an overestimate for the area of the region.. Determine an underestimate for the area of the region.. Find the actual area (Indicate expression using integral) using the graph in the graphic calculator and then algeraically using the rules for integration... Find an overestimate for the asolute area.5. Find an underestimate for the asolute area.. Find the exact value for the asolute area (Indicate expression using integral) using the graphic calculator and then algeraically using the rules for integration. EXERCISE Find the area and the asolute area of the regions descried elow: i) The region enclosed y the x-axis and the function y x x 1 on the interval,. Feel free to look at the graph first. ii) The region enclosed y y sin x, the x-axis and the lines x, x Π iii The region etween the x axis and the graph of f x x 5, on the interval, 1