Answer to Question #229405 in Chemical Engineering for Lokika

Question #229405
By Changing the order of Integration and evaluate ∫0^{a}∫0^{√a^{2}-y^{2}}(x^{2}+y^{2})dxdy `
1
Expert's answer
2021-09-08T07:47:52-0400

0a0a2y2(x2+y2)dxdyy=0y=ax=0x=a2y2(x2+y2)dxdy∫_0^{a}∫_0^{\sqrt{a^{2}-y^{2}}}(x^{2}+y^{2})dxdy \\ ∫_{y=0}^{y=a}∫_{x=0}^{x=\sqrt{a^{2}-y^{2}}}(x^{2}+y^{2})dxdy \\

From the above case, limits of integration are

x=0 to x=a2y2y=0 to y=ax=a2y2    y=a2x2x= 0 \space to \space x= \sqrt{a^2-y^2}\\ y= 0 \space to \space y= a\\ x= \sqrt{a^2-y^2} \implies y= \sqrt{a^2-x^2}\\

So new limits of integration are

x=0x=ay=0y=a2x2(x2+y2)dydx0a0a2x2(x2+y2)dydx=0a2x2x2dy+0a2x2y2dy=0a(x2a2x2+(a2x2)323)dx=0ax2a2x2dx+0a(a2x2)323dx=π16a4+πa416∫_{x=0}^{x=a}∫_{y=0}^{y=\sqrt{a^{2}-x^{2}}}(x^{2}+y^{2})dydx \\ ∫_{0}^{a}∫_{0}^{\sqrt{a^{2}-x^{2}}}(x^{2}+y^{2})dydx \\ =\int _0^{\sqrt{a^2-x^2}}x^2dy+\int _0^{\sqrt{a^2-x^2}}y^2dy\\ =\int _0^a\left(x^2\sqrt{a^2-x^2}+\frac{\left(a^2-x^2\right)^{\frac{3}{2}}}{3}\right)dx\\ =\int _0^ax^2\sqrt{a^2-x^2}dx+\int _0^a\frac{\left(a^2-x^2\right)^{\frac{3}{2}}}{3}dx\\ =\frac{\pi }{16}a^4+\frac{\pi a^4}{16}


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