Double Integrals and Surface Area

In single variable calculus, we used integrals to find the area bounded by two functions say f(x) and g(x) as  shown in the diagram below:g

 

Recall also that the area of the shaded region is given by:

 

1

From the fundamental Theorem of Calculus, f(x)- g(x)  can be re-written as:

1

 

Where y (the variable of integration)  could be any other variable apart from x.

Now putting (2) in (1) gives:

 

4

The above double integral gives the area of the shaded region.

 

Worked Example

Find to 4 decimal places the area of the path of the surface

to that lies above the disk

gb

 

Solution

Capture

 

Capture2

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