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Back to Integral Calculus

Riemann Sums

Measure a curved area with rectangles. Where each one takes its height — left, right or middle — decides how far off the sum is, and how fast it improves.

step 1 of 13

Speed

The area under f(x) = x²/2 from 0 to 3. No formula for areas of curved shapes yet — so fill it with rectangles we can measure. (The true area turns out to be 4.5.)

\text{Area} = \int_{0}^{3} \tfrac{1}{2}x^2\,dx = 4.5

\int_{0}^{3} \tfrac{1}{2}x^2\,dx = F(3) - F(0) = 4.5, \qquad |\text{error}| = 0

Error against n (log scales): left, right, midpoint

Riemann Sums

Measure a curved area with rectangles. Where each one takes its height — left, right or middle — decides how far off the sum is, and how fast it improves.

step 1 of 13

Speed

The area under f(x) = x²/2 from 0 to 3. No formula for areas of curved shapes yet — so fill it with rectangles we can measure. (The true area turns out to be 4.5.)

\text{Area} = \int_{0}^{3} \tfrac{1}{2}x^2\,dx = 4.5

\int_{0}^{3} \tfrac{1}{2}x^2\,dx = F(3) - F(0) = 4.5, \qquad |\text{error}| = 0

Error against n (log scales): left, right, midpoint

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Riemann Sums

See also

Riemann Sums

See also