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Question Number 61785 by aliesam last updated on 08/Jun/19

$$\underset{0}{\stackrel{\sqrt{3-x^2}}{\int }}\frac{xy(4-x^2-y^2)\sqrt{4-x^2-y^2}-xy}{3}dy$$

Answered by MJS last updated on 09/Jun/19

$$\int \frac{xy(4-x^2-y^2)\sqrt{4-x^2-y^2}-xy}{3}dy=$$$$=\frac{x}{3}\int y{(4-x^2-y^2)}^{3/2}dy-\frac{x}{3}\int ydy=$$$$=-\frac{x}{15}{(4-x^2-y^2)}^{5/2}-\frac{x}{6}{y}^2$$$$\Rightarrow$$$$\underset{0}{\stackrel{\sqrt{3-x^2}}{\int }}\frac{xy(4-x^2-y^2\sqrt{4-x^2-y^2}-xy}{3}dy=$$$$=\frac{x}{10}(5x^2-17+2{(4-x^2)}^{5/2})$$

Commented by aliesam last updated on 08/Jun/19

$$thankssir$$

Commented by Prithwish sen last updated on 09/Jun/19

$$\mathrm{I}\mathrm{think}\mathrm{it}\mathrm{will}-\frac{\mathrm{x}}{15}{(4-{\mathrm{x}}^2-{\mathrm{y}}^2)}^{\frac{5}{2}}$$

Commented by MJS last updated on 09/Jun/19

$${\mathrm{you}}^{\prime }\mathrm{re}\mathrm{right},\mathrm{thanks}\mathrm{for}\mathrm{checking}$$$$\mathrm{I}\mathrm{corrected}\mathrm{it}$$

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