Table 3: Midpoint estimate

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Alicia Miles
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1 The function y gx.x x 1 is shown in Figure 1. Six midpoint rectangles have been drawn between the function and the x-axis over,1 ; the areas of these six rectangles are shown in Table 1. Figure 1 Let s go ahead and calculate the Riemann sum for y g x,1 using the midpoints over six over intervals of equal width. Table has been set up to facilitate this calculation. A 1 A B B B 5 B 6 Table : Areas i Area 1 A 1. 1 A.6 B.8 B 6 5 B 6 5 Table : Midpoint estimate i 1 5 * * x i i g x x * gxi x 6 B In this example, the Riemann sum absolutely, positively, unequivocally, does not add up to something we ve seen in previous examples what is this something? 6 * g xi x : i 1 What does the Riemann Sum add up to in this example? Definite Integrals - FTC 1

2 Definition b a n * lim k f x dx f x x k xk k 1 * where the interval ab, has been partitioned in n intervals of widths xk, x k is some th point on the k interval, and the mesh size ( xk ) is the largest interval width. Let s find use our calculator s graphing feature to find x x 1 dx 1, and. 1 and. x x dx, x x dx. Let s illustrate the results on figures,, Figure Figure Figure D efinite Integrals - FTC

3 Theorem b f x dx A B a where A is the area between the curve the x-axis and B is the area between the curve Below the x-axis. y f x and the x-axis that lies Above y f x and the x-axis that lies Let s use Figure 5 to help us calculate t using the integral command on the home screen of our calculator. dt. Let s verify the result using a Riemann sum and Figure 5 Definite Integrals - FTC

4 One day, long long ago, Ginger was taking a no calculator test and wrote on her paper sin d 1. Upon receiving her graded paper, Ginger saw a note gently pointing out that she should have known that writing known that was silly? sin d 1 was a silly thing to do. Why should Ginger have Meanwhile, in the next town over, Abdou was given a multiple choice question. Specifically, Abdou, who knew absolutely no trigonometry, was told that for the function y h t shown in Figure 6, the exact value of htdt was one of the numbers 5,.5,,.5, or 5. Abdou deduced the correct answer to the question using valid reasoning. Let s recreate Abdou s correct reasoning. Figure 6 D efinite Integrals - FTC

values. Moonpie had Ginger state the indicated areas and integral values repeatedly until the proverbial light lit for Ginger.

5 The next day, Moonpie was called in to tutor Ginger. Moonpie created the following exercise to help Ginger see the strong relationship, yet different meanings, between areas and integral values. Moonpie had Ginger state the indicated areas and integral values repeatedly until the proverbial light lit for Ginger. Let s recreate Ginger s enlightenment. (Note that the curves in figures 8 and 9 are each semicircular.) A 1 = Figure 7 x dx = A 1 B 1 = B 1 xdx = y x Figure 8 A = A 1 x dx = 5 x y Figure 9 B = 1 5 x dx = x y B Definite Integrals - FTC 5

6 The Net Change Theorem If ht is the height (cm) of the water in Bahram s pool t minutes after water began to run into the pool, then h t is the rate (cm/min) at which the height was changing t minutes after water began to run into the pool. Let s suppose that water was running into the pool. a. Let s draw a sketch (Figure 1) of y ht over 7 h t dt. For clarity, let s include units in our calculation. h t was constantly 1 cm/min for the first 1 minutes that,1 and use the sketch to help us calculate 7 b. What is the contextual meaning of h t dt? c. How would we use the function h (as opposed to using the function h ) to express the change in the height from the end of the third minute of filling to the end of the seventh minute of filling? 7 d. Good golly, what s the relationship between h t dt and h7 h? 6 D efinite Integrals - FTC

7 Dimitri adopted a puppy from the pound. Little did Dimitri know, the pup was in fact quite the chow hound. If wt is the weight (gm) of the pup t days after Dimitri brought it home, then w t is the rate at which the pup s weight was changing (gm/day) t days after Dimitri brought it home. z w t is shown in Figure 9. A graph of z (gm/day) Figure 9: w a. Let s use two midpoint rectangles of equal width to 1 estimate w t dt. Let s include units in our calculation to help us understand the contextual significance of the integral value and let s explicitly state the contextual significance. t (days) b. What mathematical expression would we use involving wt (as opposed to total change in the pup s weight over the first ten days of its life with Dimitri? w t ) to express the c. What s the relationship between the mathematical expressions in parts (a) and (b) of this question? Definite Integrals - FTC 7

8 The Net Change Theorem b f x dx f b f a a Suppose that f x x x 7. a. Let s draw a sketch (Figure 5) of y f x and use the sketch to help us find f x dx. 1 b. Let s use the formula for f x to find f f 1. Voila! c. If we renamed f as g, what would we rename f? Let s restate the net change theorem using these new names. 8 D efinite Integrals - FTC

9 The Fundamental Theorem of Calculus If g is a function that is continuous over ab, and G is any antiderivative of g then: b a g x dx G b G a Let s use the FTC to evaluate each of the following integrals and interpret the results. 1 a. x dx Figure 1: y x b. htdt where ht 1 sin t 1 t Figure 11: y h t Definite Integrals - FTC 9

10 A function f is shown in Figure 1. Suppose that F is the antiderivative of f that passes through the point,1.5 Answer each question on the page in reference to these two functions. Where f is positive, F is and where f is negative, F is. Because f is always decreasing, F is always. Complete Table and use those values to help you plot F onto Figure 1. Table : Integral Values Integral Value 1 1 f xdx f xdx f f x dx 1 f x dx x dx Integral 5 6 f xdx f xdx f xdx f xdx f 1 5 x dx Value Figure 1: y f x Figure 1: y Fx What is the formula for f? Use that formula together with the known point on F to determine the formula for F and verify that y F x does indeed graph to what was drawn onto Figure 1. 1 D efinite Integrals - FTC

11 Answer each question on this page in reference to the function f shown in Figure 1.The areas of the three shaded regions are, from left to right, 15, 5, and 5. Assume that F is in reference to the specific antiderivative of f that passes through the point, 7. 7 a. Determine the value of f x dx. 5 1 b. Determine the value of f x dx. 5 c. Determine the value of F 7. Figure 1: y f x 1 d. Determine the value of 1 f x dx 7 e. Determine the value of x f x dx. 7 f. Which of the following is true about f x dx? 1 i The value is positive. ii The value is negative. iii The value is definitively zero. iv There is no way to conclude anything about the value with the given information. Definite Integrals - FTC 11

12 d t dt dx. x Let s verify the Second Fundamental Theorem of Calculus for sin t cos Let s fill in each blank correctly! d dt t 5 x dx d 8 5 t dx dt After the brakes are applied, a car decelerates at a constant rate of ft/s 15 s before coming to a complete stop 5 ft from the point where the brakes were first applied. How fast was the car moving when the brakes were first applied? 1 D efinite Integrals - FTC

The Fundamental Theorem of Calculus Using the Rule of Three

The Fundamental Theorem of Calculus Using the Rule of Three A. Approimations with Riemann sums. The area under a curve can be approimated through the use of Riemann (or rectangular) sums: n Area f ( k