Integrate-the-function-f-x-y-xy-x-2-y-2-over-the-domain-R-3-x-2-y-2-3-1-xy-4-

Question Number 98520 by bobhans last updated on 14/Jun/20

Integrate the function f (x, y) = xy (x^{2} + y^{2})

over the domain R : {- 3 ⩽ x^{2} - y^{2} ⩽ 3, 1 ⩽ xy ⩽ 4}

Answered by john santu last updated on 14/Jun/20

I = \int \int_{R} x y (x^{2} + y^{2}) d x d y

p u t x^{2} - y^{2} = u; x y = v

l i m i t s : - 3 ⩽ u ⩽ 3; 1 ⩽ v ⩽ 4 which

bounds region R

J = \frac{J (u, v)}{J (x, y)} = | \begin{matrix} \frac{\partial u}{\partial x} \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} \frac{\partial v}{\partial y} \end{matrix} | = | \begin{matrix} 2 x - 2 y \\ y x \end{matrix} |

= {2 x}^{2} + {2 y}^{2}

J (x, y) = \frac{J (u, v)}{2 (x^{2} + y^{2})}

I = \frac{1}{2} \underset{u = - 3}{\int^{u = 3}} \underset{v = 1}{\int^{v = 4}} v dvdu

{= \frac{1}{2} [3 - (- 3)) (\frac{v^{2}}{2})]}_{1}^{4}

= 3 (\frac{15}{2}) = \frac{45}{2}