Change of Order of Integration, Engineering Maths, Btech first year

Change of Order of Integration

Changing the order of Integration means: if we are integrating w.r.t. $y$ first and then w.r.t. $x$, then after changing the order we will integrate w.r.t. $x$ first and then w.r.t. $y$.
So how it is done? Simple: if we are given the problem with horizontal strip then change it to vertical strip and vice-versa. So below are the steps to follow to change the order of integration:

1) Within the problem identify the variable limits
2) From that identify whether it's horizontal strip or vertical strip
3) Trace the curve and identify the region
4) Change the strip: if horizontal then vertical and vice-versa
5) Find limits
6) Integrate

Sample Problem 1

Evaluate ${\int }_0^1{\int }_{x^2}^{x}xy(x+y)dxdy$ by changing the order of integration.

Solution

According to question,
limits of $y$ are: $$y=x^2\text{ to }y=x$$ Limits of $x$ are: $$x=0\text{ to }x=1$$ Finding points of intersection $$x^2=x\Rightarrow x^2-x=0$$ $$x(x-1)=0$$ $$\Rightarrow x=0,x=1$$ $$\text{For }x=0,y=0$$ $$\text{For }x=1,y=1$$ So the points of intersection are (0,0) and (1,1)
As you can see, $y$ has variable limits which means the strip used is vertical. So to change the order, we will use horizontal strip

After changing the order, the new limits are: $$\text{For }x,x=y\text{ to }x=\sqrt{y}$$ $$\text{For }y,y=0\text{ to }y=1$$ Now integrating $$={\int }_0^1{\int }_y^{\sqrt{y}}xy(x+y)dxdy$$ $$={\int }_0^1{\int }_y^{\sqrt{y}}(x^{2}y+x{y}^2)dxdy$$ $$={\int }_0^1{[\frac{x^{3}y}{3}+\frac{x^2{y}^2}{2}]}_y^{\sqrt{y}}dy$$ $$={\int }_0^1[\frac{y^{3/2}\times y}{3}+\frac{y^3}{2}-\frac{y^4}{3}-\frac{y^4}{2}]dy$$ $$={\int }_0^1[\frac{y^{5/2}}{3}+\frac{y^3}{2}-\frac{5y^4}{6}]dy$$ $$=[\frac{2y^{7/2}}{21}+\frac{y^4}{8}-\frac{y^6}{6}]$$ $$=[\frac{2}{21}+\frac{1}{8}-\frac{1}{6}]$$ $$=\boxed{{\displaystyle \frac{3}{56}}}$$