Mathematics-I (3110014)

BE | Semester-1   Winter-2019 | 17-01-2020

Q2) (c)

Change the order of integration and evaluate 01x1siny2 dy dx.

For, I=x=0x=1y=xy=1siny2 dy dx

Old Limit ( Fig.1 ) New Limit ( Fig.2 )
x0 to x1

yx to y1
y0 to y1

x0 to xy


Applying New Limits,

For, I=y=0y=1x=0x=ysiny2 dy dx

For,I=01siny2·x0y dy

For,I=01siny2·y-0 dy

For,I=01y·siny2 dy

Let,y2=t

2ydy=dt

ydy=dt2

Change in limits:

y=0t=0

y=1t=1

Substituting the values,

For,I=t=0t=1sin t dt2

For,I=12t=0t=1sin t dt

For,I=12-cos t01

For,I=12-cos 1+cos 0

For,I=121-cos 1

So, I=01x1siny2 dy dx=1-cos 12