MULTIPLE INTEGRAL
VECTOR CALCULUS
DIFFERENTIAL EQUATIONS
MATRICES
DIFFERENTIAL CALCULUS

Change of Order-5

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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}