How to finde the volumen between 2 Cylinders?

Hallo,

I would like to ask for your experience. I need to find the volume with triple integrals between 2 cylinders. They are not along the same axis.

To imagine how they are found, they form a cross, like this: +

The first one is an ordinary cylinder and it follows the form z= 3 sin (phi). This was really easy.

The second one is an elliptic cylinder. Here are starting the problems. The area of this second cylinder is given by x^2/a^2+y^2/b^2. For a = 2.6 and b= 3.0

But I just can transform this second one into cylindric coordinates and at the same time turn it 90° from the first cylinder to form this "+" cross.

Do you have any idea?

Thanks a lot!

Best regards.