Area under the curve without substitutions

Question

Now in general when we try to find area under the curve using integrals, we substitute the value of y in terms of x (as derived from the equation of the curve) in the function given and find the area according to given limits.

But suppose I wished to find the area under the curve without substituting any variable in terms of the other, How would I do it?

ASSUME there is a function , f(x,y)

Now I tried three ways of solving

1. The regular method of substituting y= some A(x)
   

2. By Partial Integration(not integration by parts)
   Partial Integration

3. By Double Integration
   Double Integration

The answers are different in each case.
Can someone please explain?

For example lets assume (y-2x-6)(y-3x+18)=f(x,y) and the area of curve between the points: (-4.8,-3.6) , (0,6) , (6,0) , (4.8,-3.6) is to be found

I used the following integrals:

1)Method 1 for solution
2)Method 2 for solution
3)The usual method which gives the right answer

Please Explain and provide the right method

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