doing integrals 'from first principals'

hi, I'm currently working through the calculus courses. When I learned differentiation, I enjoyed calculating derivatives by using the slope formula. e.g. instead of just using the power rule to calculate that the derivative of x^2 is 2x, I would work it out using the slope formula with increasing small values for h and watch my answer become more and more accurate as h became smaller.

Is it possible to do something similar with integrals? I tried to take the integral of x^2 between 0 and 5 using the arithmetic series formula but realized that that will only give the right answer if the function is a straight line, and x^2 is not a straight line. Is there are summation formula for integrals analagous to the slope formula for derivatives? Thanks.