Question

Reversing order of integration in f(x,y)dy do results in an equal integral of the form b [T f(x,y)dar dy. a c Determine the unknowns a,b,c,d. Which of the following is the correct value of a+c?

Answer 1

Given 1/2 Itsing frary) dy du 172x ä O from integral limits I+Sinx <y = 1+ 2x T o <x<T 2

for plotting y = 1+Sinx (Red) and y = 1+ 2x y (Green)

(元/2, 2) 2- 1.5 (0, 1) 11 -0.5 -2 0 2

Since so 2 our osast required regian is on hand side right q, y aris

(22) by=itsins 4+ 2x 지 (orl) 2

91 (1|2,2) y=sinx 8- 1+2х (10) 2 . . X

consider a horizontal Slice at y units units aboue rom x-anis. This slice from

can take value of ne from x=24 to 2=42 so " <x<H2 Put x= x, in y=1t sinx ht y = = It sina → Sinne +1=y > sinze = yol 24 = sin'cy-D

Put x=82 in y=1+2x 갱 - 며 + 242. 不 O

= 1+ 2712 a = =y 232= ㅈ y-1 X2= 117-1) 2 ง. x= x <H2 -> sin'ıy-1) <n<ly) This horizontal slice can have possible movement in vertical direction from y=1 to y=2

2. Is y=2 2/2 , It sinx 3 ma f(xy, dy dx 'It 2x 2x on leve order of ration versing integra y=2 x= 114-1) Fony) drdy yol x=sin'iy-1)

114-1) j fin,y) dx dy Sirly- with onc b comparing f(x,y) dx dy a c

a=1 b = 2 sin' (4-1) c = 14-)

oºo at C=i+sini itsinly.