Question

Parts e, f, and g only please

2. Let f(x) = -3x + 2 for 0 < x < 1. (a) If we partition the interval (0, 1) into five subintervals of equal length Ar, 0 = xo <12 <2<83 < 14 < 25 < x6 = 1, what is Ar and what are the ri? (b) Sketch a diagram for each of L5 and R5, the left and right enpoint Riemann sums for f(c) using the partition above. (c) Write in sigma notation and in expanded form) and evaluate L5 and R3, the left and right endpoint Riemann sums approximating the signed area between the x-axis and y = f(x). (d) Using basic geometry, what is the actual signed area between the x-axis and y = f(x)? (e) Now partition the interval [0,1) into n subintervals of equal length Ar = 1, so that X; = iAr = , and write left and right Riemann sums Ln, Rn, approximating the signed area between y = 0 and y = f(x) n-1 f(xi) A.r =? Rn = f(xi) A.x =? i=1 (f) Simplify the Riemann sums Ln and Rn from the previous part using the basic facts za nza 360) + ho) = 30 +316). Źc914– cŽym), constant, n=a b = c) n=a g(n), c constant, which say that you can rearrange addition and that multiplication distributes over addition, and the more complicated formula for the sum of the first N positive integers (which you might try figuring out for fun) N N(N+1) i=1+2+3+...+N = - 2 n = 2 - 3n(n-1) 2n2 2n2 3n(n +1) [You should get something equivalent to Rn = 2 - *** ., g) What happens as n +00, i.e. what are lim Rn, lim Ln? n-> n00

Answer 1

2 fm) -32+2, 2 6 (0,0) » Let un compider a partition poşno, n, or, dar of so, l] of equal lengin an follows: xion 4x = h. Osich Ln = left Riemann sum a f(xi) 4 x oń (-31 +2) - Riemann suma oň 5 3+2)

1 2n- 0. In [ 21 - (1424 --- --1)] -[2n- n(hany ] - 2-3 (1-4). and Roh [ 2n- 3 (1+2+ - - ren)] 2 - (1+6). . et un = 2-3/2012 an et Rno 2-/- 1% need