Riemann Sum Calculator

A Riemann sum is the idea that defines integration: to find the area under a curve, slice the region into thin rectangles, add up their areas, and let the slices get thinner until the total settles on the exact answer. This calculator does that for any function you give it, showing all the standard versions at once. Enter a function of x, the start and end of the interval, and how many subintervals to use, and it returns the left, right, midpoint and trapezoidal estimates of the definite integral, updating as you type. The four methods differ only in where each rectangle's height is measured. The left sum uses the value at the left edge of each slice, the right sum uses the right edge, and the midpoint sum uses the middle, which usually lands much closer to the truth. The trapezoidal rule replaces the flat tops with sloped ones by averaging the left and right heights, giving a clean, accurate estimate that often brackets the real value alongside the midpoint result. Seeing all four together is the real teaching power here, because you can watch the left and right sums sit on either side of the answer while the midpoint and trapezoidal estimates close in, and then increase the number of subintervals and see every estimate sharpen toward the same value. The calculator accepts ordinary notation, with the caret for powers and the usual sin, cos, sqrt, exp and ln, so you can integrate polynomials, roots, trig and exponential functions. That makes it genuinely useful for calculus students learning the definition of the integral, for checking numerical-integration homework, and for getting a quick area estimate when there is no neat formula. The methods and a worked example are explained clearly below.

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Use x, ^ for powers, and functions like sin, cos, tan, sqrt, exp, ln. Up to 100,000 subintervals.