Question

1. (a) Find L4 and R4 for the integral $\int ^5$ 1 (x sin x/2) dx

Show the setup and round the answer to threedecimal places.

(b) Find M4 for the integral $\int ^5$ 1 (x sin x/2) dx . Show the setup and round the answer to four decimal places.

Sketch the approximating rectangles on the graph.

(c) Compare the estimates with the actual value $\int ^5$ 1 (x sin x/2) dx $\approx$ 10.243 . Which estimate is the most accurate?

(d) Express the integral from part (a) as a limit of Riemann sums in sigma notation. Do not evaluate the limit.

$\int ^5$1 (x sin x/2) dx = lim $\Delta$ x[f(x1)+(fx2)+...+f(xn)]=lim $\Delta$ x$\sum_{I=1}^{n}f(xi)$

be sure to find $\Delta$ x= b-a/n =

Answer 1

1) ras Given integral is: S n sin ( h) dn so, Here aa 1 and 2=5, fon) = nsin (2/2) 5 - 1 (since since it is given to take 4 rectangles, Ana = n. n=4 a The left end points are: 1, 2, 3, 4 and the right end points are :2,3,4,5 1 1 1 0 5 2 3 1 4 Now, ("24) (82 (112) (4) Ms) 4 Ly= E f (res ) Ana i=1 - 1 = An [ (1)+ f(2) + +13)+ PCH) 1 [1 sin ( 22 ) + 2 sin ( 3 ) + 3 sin ( 2 ) + 4 sin (4)] = [sin ( 2 ) + 25m (2) + 3 sin ( 2 ) + 4 Sin (2)] 8.79 2 i-e How, [24 = 8.79 R4 = f (As) an an [f cha) + f (3) + f Chn) + f (Ms) = 1 [2 sin (1)+3 sin (2/2) + 4 Sin (2) + 5 sin(s) =) - 11.3049 X 11.305 i e Ru 11.30497 ñ 11.305 19 11.305

To find my fore the integreal for sm (2) de the riddle points 1 are: 1.5, 2.5, 3.5,21.5 Now, My = An [+(1.5) + f(2.5) + f (3.5) + f (4.5) (4-8) Sin(4.2)+25) Sin ( 219 ) +(3.5) sin(35) +(45) Sin (45) - 10.34019 2 10. 3402 So, = 10.340 2 4A 3 2 त 1 0 1 2 4 5 (appreoninating seretongles A1, A2, A3 & Au on the graph) C) Actual value is: sensin (nsin (2/2) on 2 10.243 Out of the three estimate Lu My and Ry My is the most accweate.

(d) we know, s fins de lim Ž f(ni) on . n> is î= 1 Ъ— С where Ana i=1,2,...,n and Ni= atian cui are the right end points) s-1 n Hi=1+1.4 b. 니 Here a=1, b=5 So, an= 5 So, a sin (42) du = lim [canit ) sin (3 4 / 3 ) sin ( h / 2 + 2il) 1 (Ang)

Plz give a thumbs up for the effort .