## Elliptical Coordinates (Double Integrals)

#### Krystian Karczyński

Founder and General Manager of eTrapez.

Graduate of Mathematics at Poznan University of Technology. Mathematics tutor with many years of experience. Creator of the first eTrapez Courses, which have gained immense popularity among students nationwide.

He lives in Szczecin, Poland. He enjoys walks in the woods, beaches and kayaking.

There are times in life when the region of integration in a double integral is an ellipse….

What do we do then?

## Elliptical Coordinates

A neat method of solving this is usually to use the so-called elliptical coordinates. It’s something like polar coordinates, the mechanism works in a similar way, only you substitute different things for x and y, and the Jacobian is different. The interpretation of ‘r’ is also different. So, to sum up, if you know how to switch to polar coordinates (which is usually done when the region of integration is a circle), you’ll easily get the hang of elliptical coordinates too.

So we have the integral: and the region of integration bounded by an ellipse centered at the origin, with the equation: . Let’s make sure the right side of the ellipse equation is 1, alright? If, for example, it’s 9, you can easily make it 1 by dividing both sides of the equation by 9.

The region of integration drawn looks like this:

What a and b mean is clear from the drawing. Be careful, because if the denominator in the ellipse equation under is, for example, 9, it means that , for obvious reasons, right?

Now with such a “clean” situation, we move to elliptical coordinates, substituting:

### Meaning of Variables in Elliptical Coordinates

The angle means exactly the same as in polar coordinates, and

### Jacobian

The Jacobian in elliptical coordinates is equal to

Remembering the Jacobian, we switch to the integral in elliptical coordinates:

where the variables

Just take it and calculate.

## Example

Calculate the integral

Following the above scheme, we substitute:

We take the region of integration:

And calculate the integral:

Which is, of course, just a formality by now 🙂

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