Find the volume of the finite region bounded by the five planes z=0, x=1, y=1, x=2, y=√3 and the surface...

...z=x/(1+(y^2))

Thanks.

Note that we can describe the solid W as:
W = {(x, y, z) | 1 ≤ x ≤ 2, 1 ≤ y ≤ √3, 0 ≤ z ≤ x/(1 + y^2)}.
(Note that z varies from the plane z = 0 to the surface z = x/(1 + y^2).)

The volume of W is given by:
V = ∫∫∫W dV.

Evaluating:
V = ∫∫∫W dV
= ∫∫∫ dz dx dy (from z=0 to x/(1 + y^2)) (from x=1 to 2) (from y=1 to √3)
= ∫∫ x/(1 + y^2) dy dx (from x=1 to 2) (from y=1 to √3)
= ∫ x dx (from x=1 to 2) ∫ 1/(1 + y^2) dy (from y=1 to √3)
= [(1/2)x^2 (evaluated from x=1 to 2)][arctan(y) (evaluated from y=1 to √3)]
= (2 - 1/2)(π/3 - π/4)
= π/8.

I hope this helps!