Integral Calculator

Definite integral

Set up the integral

Enter one real function of x and finite lower and upper bounds. The live preview shows exactly what will be evaluated.

Finite boundsReal valuesTrig uses radians

Integral result

Answer

Ready

Production-authorized result · formula 1.0.0Definite integral

2*x8.00

Signed accumulation over [1, 3] · Analytic polynomial antiderivative

Graph

Function across the entered interval

Use the curve to inspect the interval and sign changes; the numeric answer above remains the primary result.

Drawing the interval graph…

Calculation details

How the answer was obtained

MethodAnalytic polynomial antiderivative

Supported antiderivativeF(x) = x^2 + C

ConvergenceAnalytic result; Simpson cross-check passed

Numerical checks and calculation steps

Error indicator0.00

Final Simpson subintervals16.00

Simpson refinement

Successive even-subinterval estimates and the difference-based error indicator used by the bounded convergence check.

Subintervals	Estimate	Error indicator
8.00	8	—
16.00	8	0

2 rows and 3 columns.

Tokenize and parse the bounded expression without executing JavaScript source.f(x) = 2*x; trig unit = radians
Use the analytic coefficient path only for the supported polynomial subset; otherwise use bounded Simpson refinement.method = Analytic polynomial antiderivative
Preserve the entered orientation and report the unrounded signed result.integral from 1 to 3 = 8

Scope and important assumptions

The variable and result are unit-neutral. If x carries a unit, dimensional analysis and conversion are the user's responsibility; no unit is inferred or converted.
Trigonometric function arguments are radians. The expression grammar is intentionally smaller than Symbolab, Wolfram|Alpha, or a full computer algebra system.
The numeric path requires every sampled value to be finite and is intended for proper finite-interval real integrals. Improper integrals, principal values, endpoint singularities, hidden discontinuities, highly oscillatory functions, and arbitrary precision require specialist analysis.
The Simpson difference indicator is a stopping diagnostic, not a certified universal error bound. A narrow spike or discontinuity between samples can evade finite sampling.

Finite-interval accumulation

Evaluate a definite integral with its orientation and method visible

A definite integral accumulates signed contribution across a declared interval. Its meaning depends on the integrand, the bounds, and whether an analytic or numerical path produced the value.

Step-by-step guide · Finite-interval accumulation

Follow an x-squared interval through both calculation paths

Declared scope: Proper real integral on finite bounds
Polynomial path: Analytic through degree 32
Numeric path: Composite Simpson refinement

Recognize the polynomial
The analytic path constructs an antiderivative x^3/3 and evaluates F(1)−F(0), producing 1/3 before display rounding.
Refine a Simpson estimate
Composite Simpson values are computed on successively finer even subdivisions. Agreement supports the displayed cross-check; for a quadratic, Simpson's rule is exact apart from floating-point representation.
Preserve bound orientation
Reversing the bounds returns −1/3. Equal bounds return zero without manufacturing an area. These identities are part of the integral, not user-interface conventions.
Record method and diagnostic
Keep the method label, subinterval count, and error indicator with the result. A converged estimate and an analytic polynomial answer have different evidence trails even when they display the same digits.

Evaluate a definite integral with its orientation and method visible: For f(x)=x^2 on [0,1], the exact value is 1/3 and the numerical cross-check has a clear role.

Translate the accumulation question into f(x), a, and b

For a flow rate f(t), the integral from a to b represents net accumulated quantity over that time interval, provided units are coherent. For a geometric graph, it represents signed area: regions above the horizontal axis contribute positively and regions below contribute negatively. It is therefore possible for a large amount of geometric area to cancel to a small definite integral.

Enter one explicit real expression of x and two finite bounds. The parser accepts the documented arithmetic operators, constants, parentheses, implicit multiplication, and common real functions; trigonometric arguments use radians. If either bound is near a domain restriction, use the limit calculator to inspect one-sided behavior before treating the interval as proper.

Follow an x-squared interval through both calculation paths

For f(x)=x^2 on [0,1], the exact value is 1/3 and the numerical cross-check has a clear role.

Recognize the polynomial
The analytic path constructs an antiderivative x^3/3 and evaluates F(1)−F(0), producing 1/3 before display rounding.
Refine a Simpson estimate
Composite Simpson values are computed on successively finer even subdivisions. Agreement supports the displayed cross-check; for a quadratic, Simpson's rule is exact apart from floating-point representation.
Preserve bound orientation
Reversing the bounds returns −1/3. Equal bounds return zero without manufacturing an area. These identities are part of the integral, not user-interface conventions.
Record method and diagnostic
Keep the method label, subinterval count, and error indicator with the result. A converged estimate and an analytic polynomial answer have different evidence trails even when they display the same digits.

Connect slope, area, and finite sums without conflating them

Derivative relationship

An antiderivative F satisfies F′=f on the relevant interval. Use the derivative calculator to inspect a derivative at a point, but do not assume a pointwise numeric derivative proves a global antiderivative identity.

Riemann-sum relationship

The definite integral is defined through a limit of finite sums. For a polynomial sampling rule such as k², the exponent calculator can verify individual powered terms before they are totaled, while this tool evaluates the bounded continuous accumulation problem.

Geometric-area relationship

A definite integral is net signed area. If a task asks for total geometric area, split at sign changes and sum absolute regional contributions rather than taking the absolute value of a possibly canceled total.

Reject intervals that violate the numerical model

The numeric path requires every sampled value to be finite. Improper integrals, endpoint singularities, principal values, and infinite limits are outside this finite proper-integral contract.
A refinement difference is a stopping diagnostic, not a universal certified error bound. A narrow spike, unobserved discontinuity, severe cancellation, or rapid oscillation can evade a fixed sample grid.
Units are not inferred. If x is seconds and f(x) is liters per second, the result is liters; mismatched scales must be converted before entry, not repaired after integration.
The page does not compute indefinite, multiple, contour, complex, or parameterized integrals, nor does it provide a general symbolic step engine. Specialist analysis is needed when those distinctions matter.

Important limitations

Integral Calculator is limited to the finite inputs, modes, and outputs stated for the Definite integral of a real function method. Check integrand f(x), lower bound, upper bound, and refinement tolerance, validation messages, the equation, and the stated display precision before relying on definite integral, method used, antiderivative, and numerical cross-check / error indicator.

This formula passed its registered engineering tests and is published under an explicit site-owner policy authorization. No independent expert, QA, editorial, legal, or third-party review is claimed. Verify consequential results with a qualified professional.

Finite-interval accumulation

Evaluate a definite integral with its orientation and method visible

A definite integral accumulates signed contribution across a declared interval. Its meaning depends on the integrand, the bounds, and whether an analytic or numerical path produced the value.

Step-by-step guide · Finite-interval accumulation

Follow an x-squared interval through both calculation paths

Declared scope: Proper real integral on finite bounds
Polynomial path: Analytic through degree 32
Numeric path: Composite Simpson refinement

Recognize the polynomial
The analytic path constructs an antiderivative x^3/3 and evaluates F(1)−F(0), producing 1/3 before display rounding.
Refine a Simpson estimate
Composite Simpson values are computed on successively finer even subdivisions. Agreement supports the displayed cross-check; for a quadratic, Simpson's rule is exact apart from floating-point representation.
Preserve bound orientation
Reversing the bounds returns −1/3. Equal bounds return zero without manufacturing an area. These identities are part of the integral, not user-interface conventions.
Record method and diagnostic
Keep the method label, subinterval count, and error indicator with the result. A converged estimate and an analytic polynomial answer have different evidence trails even when they display the same digits.

Evaluate a definite integral with its orientation and method visible: For f(x)=x^2 on [0,1], the exact value is 1/3 and the numerical cross-check has a clear role.

Translate the accumulation question into f(x), a, and b

Follow an x-squared interval through both calculation paths

For f(x)=x^2 on [0,1], the exact value is 1/3 and the numerical cross-check has a clear role.

Recognize the polynomial
The analytic path constructs an antiderivative x^3/3 and evaluates F(1)−F(0), producing 1/3 before display rounding.
Refine a Simpson estimate
Composite Simpson values are computed on successively finer even subdivisions. Agreement supports the displayed cross-check; for a quadratic, Simpson's rule is exact apart from floating-point representation.
Preserve bound orientation
Reversing the bounds returns −1/3. Equal bounds return zero without manufacturing an area. These identities are part of the integral, not user-interface conventions.
Record method and diagnostic
Keep the method label, subinterval count, and error indicator with the result. A converged estimate and an analytic polynomial answer have different evidence trails even when they display the same digits.

Connect slope, area, and finite sums without conflating them

Derivative relationship

Riemann-sum relationship

Geometric-area relationship

Reject intervals that violate the numerical model

The numeric path requires every sampled value to be finite. Improper integrals, endpoint singularities, principal values, and infinite limits are outside this finite proper-integral contract.
A refinement difference is a stopping diagnostic, not a universal certified error bound. A narrow spike, unobserved discontinuity, severe cancellation, or rapid oscillation can evade a fixed sample grid.
Units are not inferred. If x is seconds and f(x) is liters per second, the result is liters; mismatched scales must be converted before entry, not repaired after integration.
The page does not compute indefinite, multiple, contour, complex, or parameterized integrals, nor does it provide a general symbolic step engine. Specialist analysis is needed when those distinctions matter.

Integral Calculator

Set up the integral

Answer

Function across the entered interval

How the answer was obtained

Simpson refinement

Evaluate a definite integral with its orientation and method visible

Follow an x-squared interval through both calculation paths

Recognize the polynomial

Refine a Simpson estimate

Preserve bound orientation

Record method and diagnostic

Translate the accumulation question into f(x), a, and b

Follow an x-squared interval through both calculation paths

Connect slope, area, and finite sums without conflating them

Derivative relationship

Riemann-sum relationship

Geometric-area relationship

Reject intervals that violate the numerical model

Important limitations

Set up the integral

Answer

Function across the entered interval

Integrand over the requested interval

How the answer was obtained

Simpson refinement

Evaluate a definite integral with its orientation and method visible

Follow an x-squared interval through both calculation paths

Recognize the polynomial

Refine a Simpson estimate

Preserve bound orientation

Record method and diagnostic

Translate the accumulation question into f(x), a, and b

Follow an x-squared interval through both calculation paths

Connect slope, area, and finite sums without conflating them

Derivative relationship

Riemann-sum relationship

Geometric-area relationship

Reject intervals that violate the numerical model

Integrand over the requested interval