Problem Set 6.Due in tutorial on February 1–2Instructions:• Print this cover page, fill it out entirely, sign at the bottom, and STAPLEit to the front of your problem set solutions. (You do not need to printthe questions.)Doing this correctly is worth 1 mark.• Submit your problem set ONLY in the tutorial in which you are enrolled.• Before you attempt this problem set read the notes we posted online aboutthe definition of the integral and do all the practice problems from section12.1, 5.2, 5.3 (see course website).PLEASE NOTE that so far over 20 students have been penalized for academic misconduct and now have a record with OSAI. Do not be thenext one. Re-read “Importantnotes on collaboration” on the cover page for Problem Set 1.Last name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..First name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .Student number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Tutorial code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .TA name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..Please, double-check your tutorial code on blackboard, and double-check your TA nameon the course website. Remember that if there is a discrepancy between Blackboard andROSI/ACORN, then your correct tutorial is the one on Blackboard, not on ROSI/ACORN.See http://uoft.me/137tutorialsNote: Question 0 is a warm-up question to make sure you understand the definition oflower/upper integral and the definition of supremum and infimum. Do not submit it.Only submit the other five questions.0. Let f be a bounded function on the real interval [a, b].(a) Prove that I b a(f) satisfies the following two properties:i. Lf(P) ≤ I b a(f) for every partition P of [a, b].ii. For every ε > 0, there exists a partition P of [a, b] such that I b a(f) − ε 0, there exists a partition P of [a, b] such that J − ε 0, ∃ partition P of [a, b], such that Uf(P) − Lf(P) 0, ∃ partition P of [a, b], such that Uf(P) − Lf(P) < ε.Hint: Use the definition of integrability: f is integrable on [a, b] if and only ifI ba(f) − I b a(f) = 0. Also use the definitions of I b a(f) and I b a(f). Finally, rememberthat we always know that I b a(f) − I b a(f) ≥ 0, whether f is integrable or not.3. Let f and g be two bounded functions on the interval [a, b].(a) Let P be a partition of [a, b]. Only one of the following two inequalities is alwaystrue:Lf+g(P) ≤ Lf(P) + Lg(P), Lf+g(P) ≥ Lf(P) + Lg(P)Determine which one is always true, prove it, and then show the other one isnot always true with an example.(b) Repeat Question 3a with upper sums instead of lower sums.(c) Assume that f and g are integrable on [a, b]. Prove that f + g is also integrableon [a, b].Hint: Use Problem 2 repeatedly.4. Give an example of two bounded functions f and g on an interval [a, b] such thatI ba(f + g) 6= I b a(f) + I b a(g).5. In this question, you are going to compute the exact value of Z2 5(5x − x2) dx usingRiemann sums. Let us call f(x) = 5x − x2. Since f is continuous on [1,3], we knowit is integrable. Hence, its value can be computed using any Riemann sums viaequation (5.2.7) in the book.For every natural number n, let us call Pn the partition that splits [2,5] into n equalsub-intervals. Notice that limn→∞||Pn|| = 0. Hence, we can writeZ2 5 5x − x2 dx = lim n→∞ S(Pn),where S(Pn) is any Riemann sum for f and Pn. In particular, to make things simpler,we are going to choose the Riemann sum S(Pn) where at every subinterval we usethe righ-endpoint to evaluate f.(a) Let us write Pn = {x0, x1, . . . , xn}. Find a formula for xi in terms of i and n.(b) What is the length of each sub-interval in Pn?(c) Since we are using the right-endpoint, it means we are picking x? i = xi. Useyour above answers to obtain an expression for S(Pn) in the form of a sum withsigma notation.(d) Using the formulasNX i=1i =N(N + 1)2,NX i=1i2 = N(N + 1)(2N + 1)6,NX i=1i3 = N 2(N + 1)24if needed, add up the expression you got to obtain a nice, compact formula forS(Pn) without any sums or sigma symbols.(e) Calculate limn→∞S(Pn). This number will be the exact value of Z2 5(5x − x2)dx.Hint: Your final answer should be 272.