Definite Integral Solver

Use this free definite integral solver & calculator to compute exact and numerical values of integrals. Enter your function and bounds to get the result with step-by-step antiderivative computation. Perfect for calculus students and anyone needing to calculate areas under curves.

What is a Definite Integral?

A definite integral calculates the net signed area between a function and the x-axis over a specific interval. Unlike indefinite integrals which give you a family of functions, definite integrals produce a single numerical value.

The notation \( \int_a^b f(x)\,dx \) represents the integral of \( f(x) \) from \( a \) to \( b \), where \( a \) is the lower limit and \( b \) is the upper limit of integration.

The Fundamental Theorem of Calculus

If \( F(x) \) is an antiderivative of \( f(x) \), then:

$$\int_a^b f(x)\,dx = F(b) – F(a)$$

This powerful theorem connects differentiation and integration, showing they’re inverse operations.

Signed Area

Above the x-axis

When \( f(x) > 0 \), the area contribution is positive. The region between the curve and the x-axis adds to the total.

Below the x-axis

When \( f(x) < 0 \), the area contribution is negative. This is why we call it “net signed area” rather than just area.

Properties of Definite Integrals

Additivity

$$\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$$

You can split integrals at any point.

Reversal

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

Swapping limits changes the sign.

Linearity

$$\int_a^b [af(x) + bg(x)]\,dx = a\int_a^b f(x)\,dx + b\int_a^b g(x)\,dx$$

Constants factor out, and integrals distribute over addition.

Applications

Physics

• Work done by a force
• Displacement from velocity
• Charge from current
• Energy calculations

Geometry

• Area between curves
• Volume of solids
• Arc length
• Surface area

Probability

• Expected values
• Cumulative distribution functions
• Probability density integration

Common Integrals

See more Integration Formulas.