Alternative approach in finding the solution of this Mathematical in math24 website

Here below is a link to the problem I'm concerened about. A best approach to its solution, has been provided by math24 website staff. I really like their approach!

However, I also have tried to use a different approach, to arrive at the same solution. Please see the PDF attachment below (note that in my approach, no graphical sketch is included , please refer to the original website for the same)

Geometric Applications of Double Integrals

EXAMPLE 2:

I'm sorry! The title was supposed to read:
Alternative approach in finding the solution of this Mathematical Problem in math24 website