Question

Evaluate ∫∫∫B f(x,y,z) dV for the specified function f and B:

f(x,y,z)=(z/x) 3≤x≤6,0≤y≤4,0≤z≤2

Answer 1

$$\displaystyle\int_{3}^6\displaystyle\int_{0}^4\displaystyle\int_{0}^2\dfrac{z}{x}dzdydx=$$

$$=\displaystyle\int_{3}^6\displaystyle\int_{0}^4\dfrac{1}{x}\bigg[\dfrac{z^2}{2}\bigg]{2\brack 0}dydx=$$

$$=\displaystyle\int_{3}^6\displaystyle\int_{0}^4\dfrac{2}{x}dydx=$$

$$=\displaystyle\int_{3}^6\dfrac{2}{x}\bigg[y\bigg]{4\brack 0}dx=$$

$$=\displaystyle\int_{3}^6\dfrac{8}{x}dx=$$

$$=\bigg[8\ln(|x|)\bigg]{8\brack 3}=8(\ln6-\ln3)=8\ln2$$

$$\displaystyle\int_{3}^6\displaystyle\int_{0}^4\displaystyle\int_{0}^2\dfrac{z}{x}dzdydx=8\ln2$$