Question

quanto vale essa integral?

Answer 1

Olá

Integral tripla.

$\displaystyle \mathsf{ \int\limits^1_0 \int\limits^x_0 \int\limits^y_0 {\frac{y}{x}} \, dzdydx }$

A primeira integral está em 'dz', portanto, as demais variáveis na integral, salvo a variável 'z', se tornam constantes.
Integrando em dz


$\displaystyle \mathsf{ \int\limits^1_0 \int\limits^x_0 \left[\int\limits^y_0 {\frac{y}{x}} \, dz\right]dydx }\\\\\\\\\mathsf{ \int\limits^1_0 \int\limits^x_0 \left[ {\frac{y}{x}\cdot z} \, \right]\bigg|^y_0dydx }\\\\\\\\\mathsf{ \int\limits^1_0 \int\limits^x_0 \left[ {\frac{y}{x}\cdot (y)} \, ~-~ {\frac{y}{x}\cdot (0)} \right]dydx } \\\\\\\\\mathsf{ \int\limits^1_0 \int\limits^x_0 \frac{y^2}{x} dydx }$



Agora, a segunda integral está em dy, portanto, as demais variáveis se tornam constantes, menos o 'y'.

Integrando em dy

$\displaystyle \mathsf{ \int\limits^1_0 \int\limits^x_0 \frac{y^2}{x} dydx } \\\\\\\\\mathsf{ \int\limits^1_0 \left[ \int\limits^x_0 \frac{y^2}{x} dy\right]dx } \\\\\\\\\mathsf{ \int\limits^1_0 \left[ \frac{y^3}{3x} \right]\bigg|^x_0dx } \\\\\\\\\mathsf{ \int\limits^1_0 \left[ \frac{(x)^3}{3x} ~-~ \frac{(0)^3}{3x}\right]dx }} \\\\\\\\\mathsf{ \int\limits^1_0 \frac{x^3}{3x}dx }} \\\\\\\text{Simplifica} \\\\\\\mathsf{ \int\limits^1_0 \frac{x^2}{3}dx }}$



Integrando em dx



$\displaystyle \mathsf{ \int\limits^1_0 \frac{x^2}{3}dx }} \\\\\\\\\mathsf{ \left(\frac{x^3}{9} \right)\bigg|^1_0}\\\\\\\\\mathsf{\left(\frac{1^3}{9} \right)~-~\left(\frac{0^3}{9} \right)}\\\\\\\\\boxed{\mathsf{ \frac{1}{9} }}$



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