Question

calcule a integral a seguir:

Answer 1

Pelo Teorema de Fubini, podemos resolver integrais múltiplas "de dentro pra fora", isto é, resolvemos inicialmente a integral simples mais interna e vamos passando para as mais externas. Veja, considerando I como a integral tripla dada:

$\displaystyle I=\int_0^1\int_0^x\int_0^{xy} x\,dz\,dy\,dx\\\\ I=\int_0^1\int_0^x\left[\int_0^{xy} x\,dz\right]\,dy\,dx\\\\ I=\int_0^1\int_0^x\left[x\int_0^{xy} dz\right]\,dy\,dx\\\\ I=\int_0^1\int_0^xx\left[z\right]_0^{xy}\,dy\,dx\\\\ I=\int_0^1\int_0^xx\left[xy-0\right]\,dy\,dx\\\\ I=\int_0^1\int_0^xx^2y\,dy\,dx\\\\ I=\int_0^1\left[\int_0^xx^2y\,dy\right]\,dx\\\\ I=\int_0^1\left[x^2\int_0^xy\,dy\right]\,dx\\\\ I=\int_0^1x^2\left[\dfrac{y^2}{2}\right]_0^x\,dx$

$\displaystyle I=\int_0^1x^2\left[\dfrac{x^2}{2}-\dfrac{0^2}{2}\right]\,dx\\\\ I=\int_0^1\dfrac{x^4}{2}\,dx=\dfrac{1}{2}\int_0^1 x^4\,dx\\\\ I=\dfrac{1}{2}\left[\dfrac{x^5}{5}\right]_0^1\\\\ I=\dfrac{1}{2}\left[\dfrac{1^5}{5}-\dfrac{0^5}{5}\right]\\\\ I=\dfrac{1}{2}\cdot\dfrac{1}{5}\\\\ \boxed{I=\dfrac{1}{10}}$