Question

∫∫∫e^(2x+y-z)onde G é o conjunto de pontos que satisfaz 0 < x < 1, 0 < y < ln3 e 0 < z < ln2

Answer 1

Calcular $\displaystyle\iiint_{G}{e^{2x+y-z}\,dV}$

sendo $G$ o paralelepípedo aberto formado pelos pontos

$(x,\,y,\,z)\in\;\;]0,\,1[\;\times \;]0,\,\mathrm{\ell n\,}3[\;\times \;]0,\,\mathrm{\ell n\,}2[.$


Como o sólido de integração é um paralelepípedo reto-retângulo, os extremos de integração são constantes. Portanto, podemos escolher a ordem de integração arbitrariamente. Vou escolher a ordem $dx\,dy\,dz.$

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Escrevendo a integral iterada:

$\displaystyle\iiint_{G}{e^{2x+y-z}\,dV}\\ \\ \\ =\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}\int\limits_{0}^{1}{e^{2x+y-z}\,dx\,dy\,dz}\\ \\ \\ =\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}\int\limits_{0}^{1}{e^{2x}\cdot e^{y-z}\,dx\,dy\,dz}\\ \\ \\ =\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}{e^{y-z}\cdot \left.\dfrac{e^{2x}}{2}\right|_{0}^{1}\,dy\,dz}\\ \\ \\ =\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}{e^{y-z}\cdot \left(\dfrac{e^{2\,\cdot 1}}{2}-\dfrac{e^{0}}{2} \right )\,dy\,dz}\\ \\ \\ =\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}{e^{y-z}\cdot \left(\dfrac{e^{2}-1}{2} \right )\,dy\,dz}\\ \\ \\ =\left(\dfrac{e^{2}-1}{2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}{e^{y-z}\,dy\,dz}$


$=\left(\dfrac{e^{2}-1}{2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}\int\limits_{0}^{\;\;\mathrm{\ell n\,}3}{e^{y}\cdot e^{-z}\,dy\,dz}\\ \\ \\ =\left(\dfrac{e^{2}-1}{2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}{e^{-z}\cdot (e^{y})|_{0}^{\mathrm{\ell n\,}3}\,dz}\\ \\ \\ =\left(\dfrac{e^{2}-1}{2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}{e^{-z}\cdot (e^{\mathrm{\ell n\,}3}-e^{0})\,dz}\\ \\ \\ =\left(\dfrac{e^{2}-1}{2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}{e^{-z}\cdot (3-1)\,dz}\\ \\ \\ =\left(\dfrac{e^{2}-1}{2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}{e^{-z}\cdot 2\,dz}\\ \\ \\ =\diagup\!\!\!\! 2\cdot \left(\dfrac{e^{2}-1}{\diagup\!\!\!\! 2} \right )\displaystyle\int\limits_{0}^{\mathrm{\ell n\,}2}{e^{-z}\,dz}$


$=(e^{2}-1)\cdot (-e^{-z})|_{0}^{\mathrm{\ell n\,}2}\\ \\ =(e^{2}-1)\cdot (-e^{-\mathrm{\ell n\,}2}+e^{-0})\\ \\ =(e^{2}-1)\cdot \left(-\dfrac{1}{e^{\mathrm{\ell n\,}2}}+1 \right)\\ \\ \\ =(e^{2}-1)\cdot \left(-\dfrac{1}{2}+1 \right )\\ \\ \\ =(e^{2}-1)\cdot \left(\dfrac{1}{2} \right )\\ \\ \\ =\dfrac{e^{2}-1}{2}$