Geometric Applications of Surface Integrals

Surface integrals are used for computations of

• surface area;
• volume of a solid enclosed by a surface.

Consider these applications in more details.

Surface Area

Suppose $$ be a smooth piecewise surface. The area of the surface is given by the integral

If the surface $$ is parameterized by the vector

then the surface area is

where $$ is the domain where the surface is defined.

If $$ is given explicitly by the function $$ the surface area is

where $$ is the projection of the surface $$ onto the $$-plane.

Volume of a Solid Bounded by a Closed Surface

Suppose that a solid is bounded by a smooth closed surface $$ Then the volume of the solid is given by

Solved Problems

Solution.

The surface area is given by

Here we have

Consequently,

By changing to polar coordinates, we have

Solution.

Using the spherical coordinate system, we can describe the surface of the upper hemisphere as

where $$ $$ (Figure $$).

Calculate the area element.

Find the cross product of the given vectors:

Hence, the area element is

Then the surface area of the hemisphere is given by