Question

1,2 - * - - dy (x + y) dx dy Or Of

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Solution

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( int_{0}^{a} int_{0}^{sqrt{a^{2}}-x^{2}}(x+y) d x d y ) ( =int_{x=0}^{a} int_{y=0}^{sqrt{a^{2}-x^{2}}} x d x d y+int_{x=0}^{a} int_{y=0}^{sqrt{a^{2}-x^{2}}} y d x d y ) ( =int_{0}^{a} x d x[y]_{0}^{sqrt{a^{2}-x^{2}}}+int_{x=0}^{a} d x cdotleft[frac{y^{2}}{2}right]_{0}^{sqrt{x^{2}-x^{2}}} ) ( =int_{0}^{a} x(sqrt{a^{2}-x^{2}}-0) d x+frac{1}{2} int_{0}^{a}left(a^{2}-x^{2}right) d x ) ( begin{aligned}=&-int_{a}^{0} z cdot z d z+frac{1}{2} int_{0}^{a}left(a^{2}-x^{2}right) d x =&-left(frac{z^{3}}{3}right]^{0}+frac{1}{2}left[a^{2} x-frac{x^{3}}{3}right)_{0}^{a} & begin{array}{c}10^{2} 2^{2}=a^{2}-x^{2}end{array} =& frac{a^{3}}{3}+frac{1}{2}left(a^{3}-frac{a^{3}}{3}right) end{aligned} ) ( =frac{a^{3}}{3}+frac{a^{3}}{3} ) ( =frac{2 a^{3}}{3} )