Integration is just adding up rectangles. Stack more under the curve and the Riemann sum closes in on the true area — the idea behind the definite integral.
Riemann SumsLive
approximating the definite integral
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Rectangles approximate the area under the curve. Add more and the Riemann sum converges to the true integral.
Reading this result: With n=12 midpoint rectangles the step is dx = 0.667. The midpoint rule already cancels much of the error at each rectangle. Adding subintervals shrinks the error toward the exact integral; midpoint and trapezoid rules converge fastest (error ~1/n²), so from n=12 the estimate tightens quickly as you push toward n=100.
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How it works
Riemann sums approximate the area under a curve with rectangles evaluated at their left edge, midpoint, or right edge. This tool shows the rectangles and compares the sum to a high-resolution exact value, so you can see convergence and the midpoint rule's superior accuracy.
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The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.