# Surface Integral

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Surface integral is the area integral of a scalar function or vector field over a specified surface.

This is best understood by looking at a practical problem shown below.

#### The Problem:

The area density (the mass per unit area) of this surface varies with * x* and

*, we need to find the total mass of the surface.*

**y**#### The Solution:

We can divide the surface into two-dimensional segments over each of which the area density * δ(x,y)* is approximately constant. For each segment, the density is

*, the area is*

**δ**_{i}*, and the mass for this small segment is*

**dA**_{i}*.*

**δ**_{i}dA_{i}Then the total mass will be the sum of the mass of all segments:

The smaller we make the individual segments, the closer this gets to the true mass of the whole surface.

When we make the * dA_{i}* approach zero and the number of segments

*approach infinity, this sum becomes an surface integration:*

**N**This the area integral of the scalar function δ(x,y) over the surface S. It is simply a way of adding up the contributions of little pieces of a function (the density in this case) to find a total quantity.

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