Evaluate ∫∫∫Wf(x,y,z)dV for the function f and region W specified: f(x,y,z)=48(x+y)W:y≤z≤x,0≤y≤x,0≤x≤1 ∫∫∫W(48(x+y))dV=

Answer 1

∫∫∫_Wf(x,y,z)dV = ∫₀¹∫₀^x∫_y^x48(x + y)dzdydx = 48∫₀¹∫₀^x(x + y)z_y^xdydx = 48∫₀¹∫₀^x(x² + xy - xy - y²)dydx =

48∫₀¹(x²y - y³/3)₀^xdx = 32∫₀¹x³dx = 8x⁴₀¹ = 8.

Answer 2

∫₀¹dx ∫₀^xdy ∫_y^xdz 48(x + y)

= 48 ∫₀¹dx ∫₀^xdy (x + y)(x – y)

= 48 ∫₀¹dx ∫₀^xdy (x ² – y ²)

= 48 ∫₀¹dx (x ³ – x ³/3)

= 32 ∫₀¹dx x ³

= 8