I have been given an equation describing the surface of an open paraboloid, z = a^2 - x^2 - y^2. In this case, z is the 'vertical' axis, and a is some constant. z is also greater than or equal to 0.
I was asked to calculate the volume of the paraboloid using a volume integral and have done so. When this was done, the paraboloid was closed by a 'disk' at z=0. However, considering that paraboloids are solids of revolution, I was wondering if I could represent the paraboloid in a way that allows me to find its volume, using the volume of revolution formulae.
I was thinking I could find the equation describing the paraboloid if the x or y coordinate was fixed and proceed from there, but I'm having trouble finding this equation. Any help or other advice would be appreciated.
I solved this yesterday using Matthew Pilling's method. It was much quicker than doing the original method.
The answer was Pi*a^4 all divided by 2.