Lab#7: Riemann-sums and Definite integrals using Derive- (chapter ...

Math 111 - Calculus I- Fall 2003Lab#7: Riemann-sums and Definite integrals using Derive- (chapter 5)1. Goals of this lab• To find an antiderivative or indefinite integral using Derive.• To find the definite and indefinite integrals using Derive• To find all Riemann-sums L n , R n , M n and T n using Derive• To draw all Riemann-sums L n , R n , M n and T n using Derive.• To compare the exact value of a definite integral and the Riemann-sums L n , R n and M n .• Relationship between the definite integral of a positive function and the enclosed area.• The effects of the increase or decrease of a function on the Riemann-sums L n and R ncompare to the exact value.2. Practice using new commands in Derive∫(A) Follow the steps below to find the Indefinite integral f(x) dx, i.e. the antiderivative of f(x), wheref(x) = 3x(1) Click on the button with icon ∫ , called calculate integral.(2) Enter the expression of the function (for our example, enter 3x)(3) Enter the variable (x in this case)(4) Choose ⊙ Indefinite in the integral box down to the left.(5) Enter the value of the constant if it is given or just enter c.(6) Click on OK∫.(7) The integral 3x dx will appear highlighted on the Algebra Window.(8) Click on the = button to get the answer.(B) Follow the steps below to find the Definite integral(1) Click on the button with icon ∫ .∫ 41f(x) dx, where f(x) = 3x(2) Enter the expression of the function (for our example, enter 3x)(3) Enter the variable (x in this case)(4) Choose ⊙ Definite in the integral box down to the left.(5) Then enter the Upper Limit 4 (in our example), and the Lower Limit 1 (in our example)(6) Click on OK .(7) The integral∫ 413x dx will appear highlighted on the Algebra Window.(8) Then click on the button = or ≈ to have the answer.(C) Follow the following steps to get ready to draw and to find the Riemann-sums:(1) In the Algebra Window of Derive, go to FILE / LOAD / UTILITY to open the folder UTILITY.(2) Look for a file called riemann.mth, then go to step (4). If you do not find it go to step (3).(3) Since you did not find the file riemann.mth, you should download it from the homework web page,save it ”somewhere” then load it from this ”somewhere” via the Derive Algebra Window using FILE/ LOAD (like in step (2)).(4) Highlight it and OPEN it, (it will take some time). If nothing happens, it is normal but Derive isnow ready for the riemann-sums.(5) The riemann-sums of a function f(x) are approximation of the area between the graph of f and thex-axis by n equal-width rectangular strips from x = a through x = b.(D) Follow the steps below to find L n = LEFT(f(x), x, n, a, b) and to draw it asDRAW LEFT(f(x), x, n, a, b) for f(x) = 3x, n = 8 = number of strips, a = 1 and b = 4.1

2(a) Declare the function f(x) := 3x and graph it. then go back to the algebra window.(b) Enter LEFT(f(x), x, 8, 1, 4) to have the value of Left-sum L n , then click on OK and =or ≈ .(c) Enter DRAW LEFT(f(x), x, 8, 1, 4) to draw the n rectangles for the Left-sum, click onand ≈ . Then Plot expression by clicking on —∽|— twice. If you see just dots (no rectangles)you have to go the Options/Points and choose Yes in Connect.(E) Open a new Plot Window, and do the same for R n=RIGHT(f(x), x, n, a, b) andDRAW RIGHT(f(x), x, n, a, b) to have the value of Right-sum R n , and to draw the correspondingn rectangles for n = 8 = number of strips, a = 1 and b = 4.(F) Do the same for M n= MID(f(x), x, n, a, b) and DRAW MID(f(x), x, n, a, b) to have the valueof Midpoint-sum M n , and to draw the corresponding n rectangles for n = 8 = number of strips, a = 1 andb = 4.(G) Do the same for T n= TRAP(f(x), x, n, a, b) and DRAW TRAP(f(x), x, n, a, b) to have thevalue of Trapezoid-sum T n , and to draw the corresponding n Trapezoids for n = 8 = number of strips,a = 1 and b = 4.OK3. ApplicationI. For the function f(x) = x 2 on the interval [a, b] = [0, 1] do the following, and report your resultin the given table:(1) Declare the function(2) Open a new Plot Window, graph the function, and use Set/Range from the menu to have better viewof the function on the interval [0, 1].(3) Is the function increasing or decreasing in the interval [0, 1]?∫∫1(4) Use the button to find the exact value of f(x) dx0(5) Find L n for n = 5, a = 0, b = 1 using LEFT(f(x), x, 5, 0, 1).(6) Sketch the corresponding rectangles using DRAW LEFT(f(x), x, 5, 0, 1), and print the plot window.(7) Visually, say if L 5 is an underestimate or overestimate of the exact area.(8) Repeat the previous questions (5), (6) & (7) with n = 20 to find L 20 .(9) Visually, decide which of L 20 or L 5 is a better approximation of the exact area under the graph.(10) Repeat the previous questions (2) to (9) for R n ; the right riemann-sum.II. Repeat all what you did for f(x) above to the function g(x) = 1do the following, and report your result in the given table.1+xon the interval [a, b] = [0, 1]III. Analyze the results from the table and the printed drawings, and type or write answersto the following:(1) How the increase of the values of n affects the estimations of the area by L n and R n ?(2) Is there a relationship between the increase or decrease of a function and the fact of L n or R n beingunderestimate or overestimate?(3) What can you say about the riemann-sums R n and L n compare to∫ 10f(x) dx and∫ 10g(x) dx?(4) What can you say about the value of the definite integral of a positive function and the the area betweenthe graph of f and the x-axis?