Question

Calcule a integral dupla:

Answer 1

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Solução!

$\displaystyle\int _{0}^{1}\int_{ x^{2} }^{x}xy^{2}dydx\\\\\\\\\ \displaystyle\int _{0}^{1}x \bigg[\int_{ x^{2} }^{x} y^{2} }\bigg]dx\\\\\\\\ \displaystyle\int _{0}^{1}x \bigg[ y^{2} }\bigg]_{ x^{2}}^{x} dx\\\\\\\\ \displaystyle\int _{0}^{1}x \bigg[ \frac{ y^{3} }{3} }\bigg]_{ x^{2}}^{x} dx\\\\\\\\ \dfrac{1}{3}. \displaystyle\int _{0}^{1}x \bigg[y^{3}}\bigg]_{ x^{2}}^{x} dx$


$\dfrac{1}{3}. \displaystyle\int _{0}^{1}x \bigg[(x)^{3}-( x^{2})^{3} }\bigg]dx\\\\\\\\\\\\\ \dfrac{1}{3}. \displaystyle\int _{0}^{1}x \bigg[(x^{3}-x^{6})}\bigg]dx\\\\\\\\\\\ \dfrac{1}{3}. \displaystyle\int _{0}^{1} \bigg[(x^{4}-x^{7})}\bigg]dx\\\\\\\\\\\ \dfrac{1}{3}. \bigg[(x^{4}-x^{7})}\bigg]_{0}^{1} \\\\\\\\\\\ \dfrac{1}{3}. \bigg[\dfrac{ x^{5} }{5} - \dfrac{x^{8} }{8} }\bigg]_{0}^{1} \\\\\\\\\\\ \dfrac{1}{3}. \bigg[\dfrac{ 1^{5} }{5} - \dfrac{1^{8} }{8} }\bigg]$


$\dfrac{1}{3}. \bigg[\dfrac{ 1 }{5} - \dfrac{1}{8} }\bigg] \\\\\\\\\ \dfrac{1}{3}. \bigg[\dfrac{ 8-5 }{40} }\bigg] \\\\\\\\\ \dfrac{1}{3}. \bigg[\dfrac{ 3 }{40} }\bigg] \\\\\\\\\ \bigg[\dfrac{ 3 }{120} }\bigg] \\\\\\\\\ \bigg[\dfrac{ 1 }{40} }\bigg]=0,025$


$\boxed{Resposta:~~\displaystyle\int _{0}^{1}\int_{ x^{2} }^{x}xy^{2}dydx= \frac{1}{40}=0,025}$

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