Question

Please solve all of these questions and do not skip any one of them. Also label them by 2, 3, 4a and 4b in this order.

Sketch the region and find the area of the triangle with the given vertices. . (0,0), (1,8), (4,3)

Answer 1

sol? 2. The vertices. of the triangle are (OC (48) (4:3) A (18) A-> he can't find the area of the AOAB. one time We divide this triangle (47) B 0 C in the partse 1 (0,0) 4 o Lacl as from AOAC and another is between 12 x < 4, ACB region Now, the alth of the DOAC is defined of x < 1 and 3 4x syssn [. the equation of the line passing through origin and (4,3) is and (1,8) is; y=&a y = 2 4 2 8 The equation of the straight line passing the through (118) and (4:3) is y-3 x-4 8 - 2 1-u 2 y 3 a 2-4 -3 TT y 2-3 4344-x) 4-x) = 3.420 Sa = 29 s 3 3 3 3 .. the region of integration of a ABL is, defined as 1 2 2 & 4 3 x L y L29 -sa 3 3

of . Area the AOAB - Area of a hoc t Area of & ACB dy da t dy dx 81 29 - 54 3a 3 = f (82 - 2x) dx + f ( 29 Sx - 3x) dx of a da + 8 = 291 84 29 x da 12 = 29 29 1 1 / 1 + [ 28 e 19 th 7. = 29 +2 s 29 x4 -29 46 - 29 xH6 - 2q + 29.10 2963 3 024 8 3 = 29 lib 29 58 3 3 29 +29 3 24 dhs 29 2 14.5 Twites sol: 3 Given a'yo regions are, -= 44 2 y axis When R is revolved about volume is, the * R r2) dy by Washer Method aty eb about a anis the and when revolved volume is v=lban rely) h(Y) dy, acy![by shell method

Now, R: outer radius of the disk ay r a inner radius the disk of xag3 x=44 zy 3 nay 3 and x= solving ay we get, 4y = y3 a yoo quadrant '. The region of integration in Ist Coordinate is, ozy=2 ya 72 2 i. volume when revolve about axis in, ya 2 No Al (ay) - (4372) dy f * { 1642 - 16y2 - yo ? diy It 7 16y3 - 11 > 8 7 16 3 27. a ㅋ 7 १ 13 - 2/4 = 2a. Y e2a 21 21 Now, in shell in shell method! r(y) = radim of the shell & nly) height y

(about reapis O : V = 20 y (ay. y3) dy 21 44²-ya dy = 21 [ 443 ( 25 - 2 Z y I 5 5 21 2.a 7. 2T 15 15 Now, 29 21 Z 27 15 . When volume is revolved about y asis/ the greater. Geiven, del 4.2) y = x - y = 1,0=0 V=2K by shell & method the volume ist Jº rez) n() d rela) n(n) die, as usb region of integration is, y=1 a Here York osasi te ) Ź X M(x) = 2+ [ revolving about xood and ha) = lara N = 2K 2 x 1 (nt) (1 - 12) da bx dx 2 5 x 21 I 2+ 2 4 1 1 2

출 . . t/ 2 꽃 낯부 올리 41 지호일를 통 사용에 QX 22 지 8T 15-21 55 2조쁜 tos 5 dry sol" 4.6) By Disk/washer method the volume is, (R²r2) dx 版 b 4 NK VET is ge Here the hea of integration 야이드 • R = outer radius R Inner rading 어~0 V ㅈ (1-rar da ㅈ 1-25 ta da 3 TRA + 3-2 찾기 srt- 4 ) ) بمانم 내에 ܩ 4 Ary KIN