Question

Task 3: Answer the following: a. Evaluate: Siz cos(x) sin?(x) dx (10 Marks) b. The moment of inertia, I, of a rod of mass 'm' and length 4r is given by Ar (2mx? dx where 'x' is the distance from an axis of rotation. Find I. (5 Marks) 2r Task 4: Answer the following: Using the Trapezoidal rule, find the approximate the area bounded by the curve y = ze), the x-axis and coordinates x = 0, x = 6. Consider n = 20. Also, find the accurate area of the above curve using integration and compare the areas found by Trapezoidal and analytical integration methods. (30 Marks)

Z=61

Answer 1

T a) let I - 61 Cos) si de 61 Cos(x) Sin Welse 2. 1/2 feast we will evaluate the Ondefinite integral s are cos asin asolu then apply the lumits to get the definite integral Co?) sur Cdix Cos(x) cos' W sin & Jobe Gif cos(d(1– sin?a)) s@Jaby 615 cos (a) (sin(60) (sin 6)-1) dx Is =61) orsa .

let U= sin(x) du Cose) i dx= I du cos(x) dre or du = cos adx Ja u (u ²-Dolu imela sud de 8 1 615 +6lu tc 3 fison & x) +618**) + c - 5*®)+ . I elf Cos?). sin x) dx 53 61 3 3:20 TI ya

61 Sin ² - 61 sin (x) + c 3 00:42 Va 761 GL Gon Plan ) - sa fb) – 54song 3-s 3 to 61 ()-4 (0-1) -61 + 61 3 5 15 required ansever in length = 48 mass = m J-S) " (61 ge ? ) dose X is the distance from an artis of rotation

I- 40 16/m xelx 28 - 47 Glm 20 & dix 6/M an is a anstant Since my v 3 40 61 m 27 au contents] 3 61 XL 20 3 (- (49) ² 03) GL ( 64049 61.864 xd 3 1952 sur 2. 6x8 3

area bounded by the the region carne y = 616, the x-anis and Coonellates 0-0, x=6 is green by ble ase 0 Accurate area Sobre 61 of de let U XA 2 du - are adu dx when a = 0, u = x 1-4-2 x = 6, u=8 = 2 = 3 .. Accurate ared = sobie 6le adu 122 Sole "du 122 (e") 122 (e-eº)

122, (23-1) integral con 2328.435505 D ausing Approsimate ared Trapezzoidal quale Thepexordal fulls Plates that Safecor an Age ( free) 7 8 f8c9+8f60) +2f63) +---+afran) Da = b-a. bra we have a = 0, 6 = 6, n=20 Ax= 6-0 = 3 20. Druce the interval [0,6 into n-20 Sub intervals of length. Ax=0

with the follofag endporitt a=o ,,., 22, 22,6=6 Now, we gust evaluate the Jamebion at o flers) = f(a) = f@) = 61 * 2 f@)-2 f@) =1022* = 141-7434 0 2 f@2) - 2 f) = (22* = 1646827 o a f(as) - 2 f(%) =1222* 196.934 these erelpointa 20 21091094 sum up 0 2 f(xia) - 2 662) - jaze © f(x20) = f(b) = f() = 61€ = 1228:2177 finally, fast the above values and multiply by Ngaz ie so bleu +1225.2197) 20 61+141.74397+164-6829 + 191.3341+ f2109.1095

2332.7997 Actual area 2328.4355 ban approni mate area = 2332 · 1997 ergon in approculcation - /actual approximatel 2328.4355-2332.7799 CE 4.3642 than 4 appao vilmate, area is area is grater the actual area
Thank you.