# Change of Order-5

Example 1 :

Evaluate $\int \int (x^{2} + y^{2})dy dx$ over the region bounded by $y = x^{2}, y^{2} = x$.

Solution :

The region of integration is bounded by

$C_{1} : y = x^{2}$

and $C_{2} : y^{2} = x$

The region of integration is shown in figure

$I = \int \int (x^{2} + y^{2}) dx dy$

$= \int_{0}^{1} \int_{y^{2}}^{\sqrt{y}} (x^{2} + y^{2})dx dy$

$= \int_{0}^{1} \left ( \dfrac{x^{3}}{3} + xy^{2} \right )_{y^{2}}^{\sqrt{y}} dy$

$= \int_{0}^{1} \dfrac{1}{3} (y^{3/2} + y^{6}) + (y^{5/2} - y^{4})dy$

$= \left [ \int_{0}^{1} \dfrac{2y^{5/2}}{15} - \frac{1 y^{7}}{21} + \dfrac{2}{7} y^{7/2} - \dfrac{1}{5} y^{5} \right ]_{0}^{1}$

$= \dfrac{6}{35}$