Answer to Question #101546 in Calculus for Gazzini Royal

Question

Find the volume of the solid bounded by

A. x^2 + y^2/ 2 +z = 12 in the region 0 < = x <= 2, 0< =y<=3.

B. f(x,y)=xy in the region x^2 <= y <=x and 0<=x<=1.

Answer 1

A. V=∫∫(12 -x²-(y²/2))dxdy=∫[12y-x²*y-(y³/6)]³₀=∫(36-3x²-9/2)dx = (36x-x³-9x/2) |^2₀=72-8-9=55

B. V= ∫∫xydxdy=∫[xy²] ^x_x^2 =0.5*∫(-x⁵+x³ )dx=((-(x⁶/12)+(x⁴/8))|^1₀=1/24=0.416666...