Expert: Scotto - 3/8/2010

Question
In our university mathematics course,I saw that the double integral could  give both area and volume.However,our teacher didn't give when it gave area and when volume.So could u give me some knowledge about double integral?

Answer
Most of the time a double integral is for the area and
a triple integral is for volume, but exceptions can arise.

For example, finding the amount of static in an area with a static equation
and limits in the x and y direction would be a two dimensional problem.

You can also integrate with respect to y first and have x in the limis.
Then, after that one has been done, integrate with respect to x with constant in the limits.
It can also be done vice-verse.

However, if the second problem said that there was a height of 1, then it would be a volume.
The integral would be done no differently.

Most of the time the problem tells you what is being found.

If it is in two dimensions, it is an area.  If it is in three dimensions it could be a double integral with respect to x and y and a constant heigth in z or of some function of x and integrated with respect to y and z.

Examples might be to find the volume of something with sides y = sin(x) and
y = -sin(x) and height 2.  The ends could be a 0 and π.  It might also be to
find the area underneat z = x²siny where y goes from 0 to π/2 and x goes from 3 to 5.

The main place to look is in the question to see what is being found.