The closest I can see to your set up would be

i.e. prettly close, but a minus in the y limit, and the cone inverted and dragged up z-wise by H units. I mean translated positively!

Integrating twice does get hairy. The first...

integrate sqrt(x^2 + y^2) dy - Wolfram|Alpha

is ok... click on 'show steps', also here: Integration Question

Then wrt x, integrate sqrt(R^2 - x^2) dx - Wolfram|Alpha also e.g. Finding the Integral

And integrate x^2 ln(R + sqrt(R^2 - x^2)) dx - Wolfram|Alpha

And some simple parts integrate x^2 ln x - Wolfram|Alpha and e.g. Integration of parts

Which is all going to be messy, no denying.

Maybe

... will be better.