Lebesgue integral as a Summation like Riemann

I'm an IB student who currently just finished Math 31 HL (calculus grade 12). For an Internal assessment math paper I wanted to try and derive a formula that works like the Riemann summ but rather than partioning the domain I want to partition the range similar to how the Lebsuage integral works. Now I am not familiar with number theory, and real analysis but all I aim to is derive a simple or somewhat complex formula. Do you think that this is achievable with what I know, and if so would taking a graphical approach be more benfieical to writing the equation or finding patterns.