Question

Calcule o volume abaixo de F(x,y) = 2xy e acima da região retangular R= [0,2] x [1,4] do plano 0xy

Answer 1

Para obter o volume, basta calcular o integral duplo:

$V = \displaystyle \iint\limits_R F(x,y) \textrm{ d}x\textrm{ d}y = \int\limits_0^2 \left(\int\limits_1^4 2xy\textrm{ d}y\right)\textrm{ d}x = \int\limits_0^2 x\left[y^2\right]_{y=1}^{y=4}\textrm{ d}x=$

$\displaystyle =\int\limits_0^2 x(16-1)\textrm{ d}x= 15\int\limits_0^2 x \textrm{ d}x = 15\left[\dfrac{x^2}{2}\right]_{x=0}^{x=2} = \dfrac{15}{2}(4-0) = 15\times 2 = 30.$