Example 1

Let E be the region bounded by the parabolic cylinder

y = x 2
and the planes
z = 0 and y + z = 1.

The projections of E onto the three coordinate planes are

The volume of E can be computed using any of the triple integrals

integral 1 integral 1 integral 1- y dzdy dx -1 x2 0
integral 1 integral 1-x2 integral 1-z dy dzdx -1 0 x2
integral 1 integral 1-y integral V~ y dx dz dy 0 0 - V~ y

Example 2

Let E be the region bounded by the parabolic cylinder

y = x 2
and the planes
z = 0, z = x, and y = x.

The projections of E onto the three coordinate planes are

The volume of E can be computed using any of the triple integrals

integral 1 integral x integral x dzdy dx 0 x2 0

integral 1 integral x integral x dydz dx 0 0 x2

integral 1 integral z integral V~ y integral 1 integral 1 integral V~ y dxdy dz + dxdy dz 0 z2 z 0 z y