Question

Use the Trapezoidal Rule, the Midpoint Rule, and Simpson's Rule to approximate the given integral with the specified value of n. (Round your answers to six decimal places.

pi/2 3sqrt(1 + cos(x))dr, n = 4 0

Answer 1

12 3 sqft (it cos (80).dy, m=4 1 Trapezondal Rule 3 Jul(a) +1 .da. n=4. The bapezoidal enle states that floo. dx x dx (flaco) +offe) +2 f(x) + + 2 flant) + flan)). 2 where ax=b-a he have a=0 b=n 2 n=4 4x 1/27 n=b 2. y 3n 8 8. Duude the internal 10, 2/2] into no4 subinterale of length ax = 1/8 won following endponte ato e Now, we you evaluate function at these exponte flxo) e fea) e flo) = 362. I fly) aff018) 42242640 = 612 t TL = 8-32223q 24122) -241074) = 6 12 +1 2 = 1.839371

6 2-055253 4 35193) = 24/59) - froy) = feb) = f(3) = Imally just eum wpabove valus and mus by mustry 16 =D [ 4.242640 +8.322239+ 7.839307+7-055253 16 +3 30-409509 16 = 5.977678 2 Mapoint Rule Calculator The medpont um uses medpons of subriferials (412020 (***) + f (an tany) + f(y + 2 b oda a a a teotxe) + f(y + 2) a where a xebra aran 8. Durde niternal 10,n/a] huo n=w subinterval of lengin with following endpont n =b. azon 3n 8

Noeleze finchen at there experts frete) - [ Deng) - () = 3tering til /2+) == 'poje = 0 ) - + (89) 32 Lim (20) 2, = US59054 $ [43+23) -- [ Tx+3/3) – +158) =3 cos (99) = 1 3.741675 (22+2y= + (50197 na 2+126) == 12060) = 3.379605 n 8 049999211 +4.059954 +3.741675 +3-299605) (15.3.03445) 8 6.00 6602 3 Simpson's rule. The simpson sule etare that f(x) dx Ax Ctent 4 fexe) +2m2)+447R) + 3

Date 2 fx2 + . ut affen-2) + 4 fron) + fran) where ax=b-a n & Ax=0 Duide interval (0,1/2] mto n=4 subhtervals of length are=1/8 with following enaporness Bra =b & alow, = 7.839377 2 L-12 tl 4 8 2 firo) = f(a) = f(0) = 382 4.242840. af 1) = 4 +62) = 12552 16-644478 & flar) = 2460) = 6 22 tl 9423) = 4 +130 19. 14.110507 fixy) = f(s) et (2) Finally, 29 9 D/4242640 + 16.644438 +1.839377 +14-11087 7+ ay 3 -D (4.5-837002) २५ 5.997007