
Successive Partial Integration 16-21
16.4.1 Volume of a Solid under a Surface
Let R be a region in the xy plane and let f be continuous and non-negative on R. Then, the volume of
the solid bounded below by the surface z = f(x, y) and above by R is given by
V f x y dA
R
=
∫∫
( , ) .
Similarly, the volume of the solid bounded by the region behind the function y = f(x, z) and in front
of the region R in the xz plane is given by
V f x z dA
R
=
∫∫
( , ) .
And the volume of the region bounded behind the function x = f(y, z) and in front of the region R
in the yz plane is given by
V f y z dA
R
=
∫∫
( , ) .
Example 16.22 Find the volume of the solid bounded ...