Question

1. Compute the following integrals: 9 (a) S (x+y+2)dA where T C R2 is the triangle with vertices (-1,-1), (0, 2) and (1,1) (b) Sp (3x + 6y)<dA where D is the quadrilateral with vertices (0,1), (2,0), (0, -1) and (-2,0) 2 9

Answer 1

1 We have to compute the integrals. a) Given Sext (x+y+2) dA where I CR² is the triangle with T Vertices (-1;1), (0,2) and (1,1). draw the triangle. Let A (-1,-1), B(0,2) and C (1,1). We bave AABC first let B (0,2) y=-X+2 (1,1) с + -1 (-1,-1) A Let us hind of a line conneching sides. We each the equation of know that the equation two points (x,y) and (72, 9₂) is y-9, (92-9) (x-x) (12-) let us Using this equation hind the equahon Of Rides. Line joining A(-;-)) and B(3) is 2--1 Y-E0) (x-60) → Line Y+1 3(x+1) y = 3x+2. joining B 104) and eat is 9-2-4-2)(-0) ) 4-2--* >> yo -442 joining coil and A (-4,-1) is 9-1-1-1 (2-0) > y=x 74 y Line

If we limit move the vertical line L from x=-1 to no, your gy Vanes from y= 3x+2 .; and further if we move the vertical line L from a=0 boxal, we have y=x to Our 9 limits as y = x to y = -x+2. this we cover Our region triangle By doing 0 3x+2 1-x+2 : j exty + 2)dA - JS (x +y +2) drydy + s (exty +2) duydge X 0 -x+2 Slaytury 2y) de + (xy + y² + 2y) dy x 2 0 $(213 4(34+2) + (3x+2)*+ 2(34+2) - (x++ x + 2x)) dx +f4(+242) + [x+2)2(-x+2) - (me?+ x + 2x)) de O Sleri 2x+4 (92°+ 12x+++62+4 3 x°- 2x]dx +|+*+22 44(2=48+4) – 22+4* Bet*2de 0 = $(6x<+12x +6]dx + 562x2-4x + 6 )dx •( 6 x 120 fox -2x + 4x² + 6x २ =0-(-276-6)+(***+6) = 2 + 10 2+ 16 3

(b) Given $(3x+by} dA where Da the quadrilateral with Vernices (0,1),(2,01; (0,01) and (-2,0) Let us draw the quadrilateral. bet A (0,1), B(2,0), ((0,-1), D(-2,0) be the points. NO) x + y (4+2) D(-2,0) 662,0) Y = -1 - - - Ciort) ry 14-2) 8 As the X Y 0-1 3-0 Eqn of line done above pooblem, let us hind the equanons of each line segments. Eqn of line joining A.CO.I) and Blag is Y-1 .(x-o) s) y 1 joining B Carol and c 4-0 = -(x-2) a) y = /(x-2) line joining clo, i and D2, og is O-- y-(1) - (4-0). =) Y+1=» y -2-0 line joining D(-2,0) and AOI) y-o-1-0 (4--2) > y = (x+2) -1-0 0-2 Eqn of Egn of If L from x=-2 to 20 your y we move the vertical line limits will be y = - 2 - 3 - 1 to y= 1 / 1842), and further if we move L from will be vertical line the Y-Olo x=2 your y limit y = $(x-2) ho you +1.

2 0 2(x+2) ht {(3x+by} dA = SJ (3x+6y} dy dx + IS (3x+ by) dy de ☺ D 2 0 (x-2) 0 {(x+2) 21x+2) Now S S (34+by} dy du 5 (93 9**+ 36 xy + 36y3).dy dx ---- -2 २ S (9x²y + 18 xy² 12ysje dx 0. Scak (5 (2+2) + {+") + 18x ((142)7(6+)*)+12 (4212)*(+4)*)dx. 2 2 9x 2 9x²9x² +9x 18x ( x + 4x + 4 4 4x + 4 2761+122+8). + 12(+*+62+12 Hox + 국 x 2 4 4

j(9_ *+182*+1876 OJ + 3(2}+6x#12x+8)) dx = $(9x+18x'+ 324 1827 36x+24) dx 2 12 x° + 36x +36x+24) dx (3x4+./2x® +18x1.24x) -2 Fox (48696 + 72 - 48) +24 a Now 2 112+2) (3x+6yj dy dx = ( ( (98²4 36x + 364²) dy die + O -7 - Creolen ise Jeteze (2+2) ( (9x²y + 18 x y² + 1243) abyde -743) 2 SC942 (1:2 + (843) + 18x (432) +- (64253). + 18x (62)*- (09:23+ 12 (Cx2). (422)*)d« 51924*+2) + 18x80 + 12 (1872))) 2 E) dx [(97°+187+ +3 ( x+6x?+ 12x + 8 8)) dx. 2 j izv?i 36x?t 367+24) dx = (3x *f12x® + 18x7 292) 8 [ 48 +96+72+48] 264

Peet and in O we get S (3x+6y) da 24 +264 288 10
using the idea of double integration we have solved the problem