O edcc.instructure. XO Acing Google Co xO Kevawn «Redund xM Whole Course Re XM Invitation: CS&e PapagoC Get Homework E XGet HomeworkA variables2 - Solve XA WAMAP - Posts x5.2 Calc.pdf+A https://wamaps3.s3.amazonaws.com/ufiles/222249/5.2_Calc.pdfThese are equal by the distributive property. The-1other formulas are discussed in Appendix F.11(а, — b) —E b.a, -EXAMPLE 2 Evaluating an integral as a limit of Riemann sums(a) Evaluate the Riemann sum for f(x) = x' – 6.x, taking the sample points to be rightendpoints and a = 0, b = 3, and n = 6.(b) Evaluate (:x' - 6x) dx.SOLUTION(a) With n = 6 the interval width isb - a3 - 0Ar =%3D62and the right endpoints are x, = 0.5, r2 = 1.0, x; = 1.5, x, = 2.0, xs = 2.5, andX, = 3.0. So the Riemann sum isR, =i-= f(0.5) Ar + f(1.0) Ar + f(1.5) Ar + f(2.0) Ax + f(2.5) Ax + f(3.0) Ax5y =x'- 6x= }(-2,875 – 5 – 5.625 – 4 + 0.625 + 9)= -3.9375Notice that f is not a positive function and so the Riemann sum does not represent a sumof areas of rectangles. But it does represent the sum of the areas of the blue rectangles(above the x-axis) minus the sum of the areas of the gold rectangles (below the x-axis) inFIGURE 5Figure 5.347SECTION 5.2 THE DEFINITE INTEGRAL(b) With n subintervals we haveb - a3Thus Xp = 0, x, = 3/n, x, = 6/n, X= 9/n, and, in general, x, = 3i/n. Since we areusing right endpoints, we can use Theorem 4:3i 3[(r' - 6x) dx = lim E S(x,) Ax = limte sum, N is a constant (unlikenve 3n in front of the E signso we can= limEquation 9 withe오후 12:57P Type here to search^O G ) O2021-01-07 See Example 2 in Section 5.2. Suppose that in part (b) we change the upper limit of integration to 2,resulting in| (2 – 6z)dr. To find the value of this integral, work through the following step-by-stepprocess.a) We know that a =and b =b) Using n subintervals, Ar =c) Assume that the sample points in each interval are right endpoints. Find the following:21 = a + Az =22 = a + 2Az =23 = a + 3Az =In general, the ith sample point is r; = a + iAr =Note: simplify your answer for thisquestion!d) Now find the sum of the areas of n approximating rectangles. Note: your answer will be an expressionin terms of n only. You really need to use the information in example 2 to help you complete this question!f(r:)Az =i=1e) Finally, find the value of the integral by letting the number of rectangles approach infinity.[ (2" – 6z)dz = lim E f(#:)Az =1=1Question Help: O Message instructor D Post to forumSubmit Question

As per Bartleby guidelines, we are allowed to solve 3 parts of a question. Given ∫02(x3-6x)dx

O edcc.instructure X O Acing Google Co x O Kevawn Redund x M Whole Course Re x M Invitation: CS& x e Papago C Get Homework X C Get Homework X O variables2 - Solv x WAMAP - Posts X O 52 Calc.paf i https://wamaps3.s3.amazonaws.com/ufiles/222249/5.2 Calc.pdf These are equal by the datrbutve property The other formulas are discussed in Acpends 11 È la, - b.) = E a, – É b, EXAMPLE 2 Evaluating an integral as a limit of Riemann sums (a) Evaluate the Riemann sum for f(x) -x' - 6x, taking the sample points to be right endpoints and a - 0, b-3, and a -6. (b) Evaluate '(r' - 6x) dx. SOLUTION (a) With a-6 the interval width is -a3-0 Ar= Ar- 6 and the right endpoints are x, - 0.5. x: = 1.0, x = 1.5, x. - 2.0. xs = 25, and X - 3.0. So the Riemann sum is R, - E fLr,) Ar - S(0.5) Ar + S(1.0) ar + f(1.5) Ar + f(2.0) Ar + f(2.5) Ar + f(3.0) Ax y'-f -(-2.875 - 5 - 5.625 - 4 + 0.625 + 9) - -3.9375 Notice that f is not a positive function and so the Riemann sum does not represent a sum of areas of rectangles. But it does represent the sum of the areas of the blue rectangles (above the x-axis) minus the sum of the areas of the gold rectangles (below the x-aisi in Figure 5. FIGURE S 347 SECTION S2 HE DEFINITE INIEGRAL th) With a suhintervals we have Thus X0, x - yn, x;- 6/n, , 9/n, and, in general, x,- 3i/n. Since we are using right endpoints, we can use Theorem 4: C - 6z) dr = lim E fix.) Ar - lim a Constant unike il so we can ientont of the sign - lim - P Type here to search O Ei A O A 40 ® 한 2 12:57 局 2021-01-07

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...