MathCalcLab

Definite Integral Calculator

Calculate definite integrals numerically with the trapezoidal or Simpson's rule and see the signed area under the curve

Enter function using x as variable. Supported: +, -, *, /, ^, sin, cos, tan, exp, ln, log (base 10), sqrt, abs

Results

Enter values and click Calculate to see the result.

Theory & Formula

Theory

A definite integral ∫[a,b] f(x) dx represents the signed area between the curve f(x) and the x-axis from x=a to x=b. It has applications in physics (work, distance), probability (cumulative distribution), and many other fields.

Fundamental Theorem of Calculus

\(\int_a^b f(x) \, dx = F(b) - F(a)\)

Where F(x) is any antiderivative of f(x). This connects differentiation and integration.

Trapezoidal Rule

\(\int_a^b f(x) \, dx \approx \frac{h}{2} \left[ f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b) \right]\)

Approximates the area using trapezoids. More accurate with larger n.

Simpson's Rule

\(\int_a^b f(x) \, dx \approx \frac{h}{3} \left[ f(a) + 4\sum_{i=1,3,5}^{n-1} f(x_i) + 2\sum_{i=2,4,6}^{n-2} f(x_i) + f(b) \right]\)

Approximates using parabolic arcs. Generally more accurate than trapezoidal rule.

Example

Calculate ∫[0,2] x² dx using the fundamental theorem:

\(\int_0^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^2 = \frac{8}{3} - 0 = 2.667\)

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