Question

Consider the function f(x)=x22−9.

(1 point) Consider the function f(x) = 9. 2 In this problem you will calculate " ( - ) dx by using the definition Lira f(x) dx = lim f(x;)Ar i=1 The summation inside the brackets is R, which is the Riemann sum where the sample points are chosen to be the right-hand endpoints of each sub- interval. r2 Calculate R, for f(x) = -9 on the interval [0, 3] and write your answer as a function of n without any summation signs. You will need the 2 summation formulas on page 369 of your textbook. We recall a few of those formulas here: n Σ i= n(n + 1) 2 i=1 Σ. n(n + 1)(2n + 1) 6 i=1 1 (ca; + b) = 0 ($)+() i=1 3 Hint: Xi = 3i and Ax = n n R, = lim Rx = n00

Answer 1

² Here f(0) = -g. 2 X; = 31_and ar = 3 :- RA To find Rnie I f(xi) 4x and lim Rp j=1 h Rp = E f (I;) AX 2 i=1 To find foxi) put x= x; in o :. f(xi) (311) 9 g X, 2 2 2 i, f(xi) fgi2/12 9- 2 f(x) = 992 .g. 2n2 n 9 Rn gi i=12n2 Σ Ahn 27 271 203 i=1 ... R = 27 2n3 il i 2 27 1=1

i', 27 n(n) (2n+1) 6 27 (A) 2n3 h Rn 27 [nent) (2n+i)] 12 n3 27 Now to find lim RA no lim Rn he lim no 27] 27 n3 (1+1^)(2+n) -27 -27 27 [ n(n+1)(2+1) 7 12n3 Tim Rn lim hoo nto 12 n3 ..lim n Rn lim nto 27 (1+y)(2+ yo) 12 - 27 :. lim Rn no 2700 (2) 12 - 27 Slim RA g 2 27 9-54 - 45 nyoo lim Rn= - 45 nx 2 ' Answer RA= 27 [ n(n+1)(207)] 12n3 27 and lim Ro - -45 nyo 2