I share your doubts, but if mathematics doesn't forsake me, it has to be 
right.

When a curve f(x) crosses the x-axis at point t, for small distances close 
to t, the curve approximates a straight line, right?  NO, IT DOESN'T. 
That's one flaw in my method.  If I take the derivative of f(x) at t, the 
tangent line is a straight line.

One area is between f(x) and the x-axis from t to (t + delta t), and the 
other area is between f(x) and the x-axis from (t - delta t) to t.  I simply 
say that when f(x) crosses the x-axis at f(x)=0, THE TWO AREAS CANCEL TO 
ZERO.

A better strategy is to use the areas above and below the tangent line when 
f(x) crosses the x-axis.

g(x) = f '(x) x + b  equation of tangent line

INT (f '(x) x + b) dx  from (t - delta t) to (t + delta t) = 0

For instance, using the example suggested by Fran