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VisualizationsCalculus & Optimization

Integrals as Area

Slide to fill the area under a curve with more rectangles. The thinner they get, the closer their total comes to the true area.

Filling the Area with Rectangles

Rectangles:4

f(x) = x² on the stretch from 0 to 3 (estimate = 5.906)

01230369
Rectangle total: 5.906
True area: 9.000
Gap: 3.094

Each rectangle takes its height from the left edge of its slice. On a rising curve it stays under the true area. Drag the slider up and the gap shrinks toward zero. That shrinking gap is what an integral is.

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