Is there any way to simplify the Area function?

[urban gyro]

I got inspired by @cerulean lily's work on analytical disk motion blur, but here’s the deal: I’m integrating u(x, y, t) / (b - a) from t = a to t = b. My approach involves turning the inequality condition in u into equality, solving for t, and getting two sets of solutions: A(x, y) + B(x, y, k) and -A(x, y) + B(x, y, k) (the first is larger)

Then, I compute n1 as the smallest k where the larger solution is > a, and n2 as the largest k where the smaller solution is < b. After that, I sum the areas for all intervals in between, excluding n1 and n2, and add the partial areas at n1 and n2. Finally, I divide everything by (b - a) and simplify