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 \(\hspace{5pt}\large \int^1_0 \int^x_{x^2} xy(x + y)dxdy \hspace{5pt}\) 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: \[ \text{For } x, \hspace{5pt} x = y \text{ to } x = \sqrt{y} \] \[ \text{For } y, \hspace{5pt} y = 0 \text{ to } y = 1 \] Now integrating \[ = \int^1_0 \int^{\sqrt{y}}_y xy(x + y)dxdy \] \[ = \int^1_0 \int^{\sqrt{y}}_y (x^2y + xy^2)dxdy \] \[ = \int^1_0 \left[\frac{x^3y}{3} + \frac{x^2y^2}{2} \right]^{\sqrt{y}}_y dy \] \[ = \int^1_0 \left[\frac{y^{3/2}\times y}{3} + \frac{y^3}{2} - \frac{y^4}{3} - \frac{y^4}{2} \right] dy \] \[ = \int^1_0 \left[\frac{y^{5/2}}{3} + \frac{y^3}{2} - \frac{5y^4}{6} \right]dy \] \[ = \left[\frac{2y^{7/2}}{21} + \frac{y^4}{8} - \frac{y^6}{6} \right] \] \[ = \left[\frac{2}{21} + \frac{1}{8} - \frac{1}{6} \right] \] \[ = \boxed{\frac{3}{56}} \]
