calcule a integral a seguir:
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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](https://tex.z-dn.net/?f=%5Cdisplaystyle%0AI%3D%5Cint_0%5E1%5Cint_0%5Ex%5Cint_0%5E%7Bxy%7D+x%5C%2Cdz%5C%2Cdy%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cint_0%5Ex%5Cleft%5B%5Cint_0%5E%7Bxy%7D+x%5C%2Cdz%5Cright%5D%5C%2Cdy%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cint_0%5Ex%5Cleft%5Bx%5Cint_0%5E%7Bxy%7D+dz%5Cright%5D%5C%2Cdy%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cint_0%5Exx%5Cleft%5Bz%5Cright%5D_0%5E%7Bxy%7D%5C%2Cdy%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cint_0%5Exx%5Cleft%5Bxy-0%5Cright%5D%5C%2Cdy%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cint_0%5Exx%5E2y%5C%2Cdy%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cleft%5B%5Cint_0%5Exx%5E2y%5C%2Cdy%5Cright%5D%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cleft%5Bx%5E2%5Cint_0%5Exy%5C%2Cdy%5Cright%5D%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1x%5E2%5Cleft%5B%5Cdfrac%7By%5E2%7D%7B2%7D%5Cright%5D_0%5Ex%5C%2Cdx "\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}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%0AI%3D%5Cint_0%5E1x%5E2%5Cleft%5B%5Cdfrac%7Bx%5E2%7D%7B2%7D-%5Cdfrac%7B0%5E2%7D%7B2%7D%5Cright%5D%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cint_0%5E1%5Cdfrac%7Bx%5E4%7D%7B2%7D%5C%2Cdx%3D%5Cdfrac%7B1%7D%7B2%7D%5Cint_0%5E1+x%5E4%5C%2Cdx%5C%5C%5C%5C%0AI%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cdfrac%7Bx%5E5%7D%7B5%7D%5Cright%5D_0%5E1%5C%5C%5C%5C%0AI%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cdfrac%7B1%5E5%7D%7B5%7D-%5Cdfrac%7B0%5E5%7D%7B5%7D%5Cright%5D%5C%5C%5C%5C%0AI%3D%5Cdfrac%7B1%7D%7B2%7D%5Ccdot%5Cdfrac%7B1%7D%7B5%7D%5C%5C%5C%5C%0A%5Cboxed%7BI%3D%5Cdfrac%7B1%7D%7B10%7D%7D "\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}}")
EFSkinha:
muito obrigado... me ajudou muito
De nada!
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