∞Limitless

CALCULUS · INTEGRALS

Area & Riemann Sums

Approximate the area under a curve with rectangles, then let them get infinitely thin.

14 min read 3 practice questions
To find the area under f(x)f(x) from aa to bb, slice the interval into nn rectangles of width Δx=b−an\Delta x = \frac{b - a}{n}. Add up their areas to get a Riemann sum.
∫abf(x) dx=lim⁡n→∞∑i=1nf(xi∗) Δx\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x

The definite integral is the limit of Riemann sums

Riemann sum builderInteractive
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Approximate the area under f(x) = x² on [0, 4]. Add rectangles and compare sampling methods.

Sample point

Approximation14.000

Exact 643\tfrac{64}{3}21.333

Error-7.333

Worked example · A left Riemann sum

Estimate ∫04x2 dx\int_0^4 x^2\,dx using 4 left rectangles.
  1. 1
    Δx=1\Delta x = 1. Left endpoints: 0,1,2,30, 1, 2, 3.
  2. 2
    Heights: 0,1,4,90, 1, 4, 9.
  3. 3
    Sum: (0+1+4+9)(1)=14(0 + 1 + 4 + 9)(1) = 14.
  4. 4
    The exact area is 643≈21.33\frac{64}{3} \approx 21.33 — left sums underestimate an increasing function.

CHECK YOUR UNDERSTANDING

Practice

Answer each question, then check.

1
Estimate ∫04x2 dx\int_0^4 x^2\,dx with 4 right rectangles.
2
What is Δx\Delta x when dividing [2,8][2, 8] into 33 subintervals?
3
For a function that is decreasing on [a,b][a, b], a left Riemann sum will…

Numeric answers accept decimals or fractions like 3/4.