Integral Calculator
Compute definite integrals numerically using Simpson's rule. Evaluate area under any curve f(x) between bounds with plot. Free online integral calculator.
About This Calculator
This Integral Calculator computes definite integrals numerically using Simpson's rule, a powerful numerical integration technique. Enter any function f(x) along with lower and upper bounds to evaluate the signed area under the curve between those limits. The result represents the accumulated value of the function over the interval [a, b].
The calculator uses Simpson's rule with 200 subintervals to achieve high accuracy for smooth functions. Simpson's rule approximates the integral by fitting quadratic polynomials through pairs of subintervals, offering O(h⁴) convergence — significantly more accurate than the trapezoidal rule for smooth functions. It also displays an interactive plot showing the area under the curve between the specified bounds, helping visualize the integral as the region between f(x) and the x-axis.
Methodology: The definite integral ∫ₐᵇ f(x) dx is approximated by dividing [a, b] into n even subintervals of width h = (b - a)/n. Simpson's composite rule applies the formula (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f(xₙ₋₂) + 4f(xₙ₋₁) + f(xₙ)], where n must be even. For polynomial functions up to degree 3, Simpson's rule gives exact results. The calculator supports a wide range of mathematical expressions including trigonometric, exponential, logarithmic, and polynomial functions.
Applications: Definite integrals are used extensively across science and engineering. In physics, they calculate work done by a variable force, center of mass, moment of inertia, and electric flux. In probability, integrals of probability density functions give cumulative probabilities. In economics, integrals compute consumer and producer surplus. In engineering, they determine area moments, fluid pressure forces, and signal energy. Students in calculus courses use integrals to find areas between curves, volumes of solids of revolution, arc lengths, and surface areas.
India (IN): Integration is covered in CBSE Class 12 mathematics and is essential for JEE Advanced. Students learn indefinite and definite integrals, integration by substitution, integration by parts, and applications like area under curves and differential equations.
United States (US): Integrals are a core topic in AP Calculus AB and BC. Students learn Riemann sums, the Fundamental Theorem of Calculus, substitution, integration by parts (BC), and applications including area, volume, and average value.
United Kingdom (UK): A-Level Mathematics covers integration in Pure Mathematics. Students learn indefinite and definite integration, integration of standard functions, substitution, and integration by parts. Applications include areas between curves and volumes of revolution.
Frequently Asked Questions
How does the integral calculator work?
The calculator uses Simpson's rule, a numerical integration method that approximates the definite integral by fitting quadratic curves through intervals of the function. It divides the area under the curve into 200 subintervals by default and applies the Simpson formula: integralᵇₐ f(x) dx ≈ (h/3)(f0 + 4f1 + 2f2 + 4f3 + ... + fₙ).
What is a definite integral?
A definite integral integralᵇₐ f(x) dx represents the signed area under the curve f(x) between x = a and x = b. It is defined as the limit of Riemann sums as the partition width approaches zero. The Fundamental Theorem of Calculus connects definite integrals to antiderivatives.
What is Simpson's rule?
Simpson's rule is a numerical integration method that approximates integralᵇₐ f(x) dx by dividing [a,b] into n even intervals and fitting a quadratic through each pair of intervals. It uses the formula (h/3)(f0 + 4f1 + 2f2 + 4f3 + ... + fₙ). For smooth functions, it has O(h^4) accuracy, making it much more accurate than the trapezoidal rule.
What mathematical expressions are supported?
Supported operations: +, -, *, /, ^ (exponentiation), sin(), cos(), tan(), asin(), acos(), atan(), sqrt(), abs(), log()/ln(), exp(), and constants pi and e. Enter expressions like 'x^2 + 3*x' or 'sin(x)'.
What is the difference between definite and indefinite integrals?
A definite integral integralᵇₐ f(x) dx gives a numerical value representing the area under the curve between bounds a and b. An indefinite integral integral f(x) dx gives a family of antiderivative functions F(x) + C, where F'(x) = f(x). The Fundamental Theorem of Calculus links them: integralᵇₐ f(x) dx = F(b) - F(a).
What are the basic integration rules?
Key integration rules: power rule (integral xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1), constant multiple rule (integral cf(x) dx = cintegral f(x) dx), sum rule (integral (f+g) dx = integral f dx + integral g dx), integration by parts (integral u dv = uv - integral v du), and substitution (integral f(g(x))g'(x) dx = integral f(u) du where u = g(x)).
How accurate is Simpson's rule for numerical integration?
Simpson's rule has O(h⁴) error, meaning the error decreases as the fourth power of the step size. For smooth functions (four times continuously differentiable), doubling the number of subintervals reduces the error by a factor of 16. This makes Simpson's rule significantly more accurate than the trapezoidal rule, which has O(h²) error. With 200 subintervals, the calculator achieves high precision for most practical purposes.
What functions can I integrate with this calculator?
The calculator supports a wide range of mathematical functions: polynomials (like x^2 + 3x - 5), trigonometric functions (sin(x), cos(x), tan(x)), inverse trigonometric functions (asin(x), acos(x), atan(x)), exponential and logarithmic functions (exp(x), log(x), ln(x)), and combinations using addition, subtraction, multiplication, division, and exponentiation (^). Constants pi and e are also recognized. The function must be defined and finite across the entire integration interval.