## Volume Bounded by a Surface and a Region in the Plane

Suppose we have a surface
$z=f(x,y$
where
$z$
is the height of the surface above the
$xy$
plane.

The volume bounded by the surface and the region
$R$
in the
$xy$
plane is
$V = \int_R f(x,y) dx dy$
.
Example: Find the volume bounded by the surface
$z=x^3 + y^3$
and the triangle in the
$xy$
plane between the coordinates
$(0,0,0), ((1,1,0), (1,0,0)$

The triangle has boundaries
$x=0, x=1, y=x$
so we can write as the region of integration as
$0 \leq x \leq 1 , y \leq x$

The integral becomes
\begin{aligned} V &= \int^1_0 \int^x_0 (x^3 +y^3) dy dx \\ &= \int^1_0 [x^3 y + y^4/4]^x_0 dx \\ &= \int^1_0 5x^4/4 dx \\ &= [x^5/4]^1_0 \\ &= 1/4 \end{aligned}