From Speedometer to Distance
Your speedometer reports velocity moment by moment, but the odometer reports total distance. The mathematical bridge between a rate and its total is the definite integral: it accumulates the values of a function between two bounds into a single number. That number might be an area, a distance, a total cost, or a probability. The Definite Integral Calculator evaluates integrals from a to b using the fundamental theorem of calculus, showing the antiderivative and the bound substitution.
The Formula, Up Close
Formula: integral from a to b of f(x) dx = F(b) - F(a)
In plain words: find any antiderivative F of f, meaning a function whose derivative is f, then evaluate F at the upper bound b and subtract F at the lower bound a. The result is the signed area: regions above the x-axis count positive and regions below count negative. This is the fundamental theorem of calculus, the link between differentiation and integration. For the indefinite version without bounds, see our indefinite integral calculator.
How to Use It
- Enter the integrand f(x), for example 2x + 1.
- Enter the lower bound a, for example 1, and the upper bound b, for example 4.
- The calculator finds an antiderivative F(x).
- It computes F(b) - F(a) and reports the value with the substitution shown.
Worked Example, Step by Step
Evaluate the integral from 1 to 4 of (2x + 1) dx.
Step 1: Antiderivative. The antiderivative of 2x is x^2 and of 1 is x, so F(x) = x^2 + x. Step 2: Evaluate at the upper bound: F(4) = 16 + 4 = 20. Step 3: Evaluate at the lower bound: F(1) = 1 + 1 = 2. Step 4: Subtract: 20 - 2 = 18.
Geometric check: the region under y = 2x + 1 from x = 1 to x = 4 is a trapezoid with heights 3 and 9 and width 3, giving area (3 + 9) / 2 x 3 = 18. The geometry confirms the calculus.
Where Definite Integrals Work Hard
Physicists integrate force over distance to get work and velocity over time to get displacement. Economists integrate marginal cost to get total cost. Probabilities are areas under density curves. Numerical methods like the trapezoidal rule approximate the same values when no elementary antiderivative exists, and the improper integral calculator extends the idea to infinite bounds.
Mistakes Worth Avoiding
- Subtracting in the wrong order. It is F(b) minus F(a); reversing gives the negative of the correct answer.
- Forgetting the lower bound. Evaluating only at b misses half the theorem.
- Ignoring the sign. Area below the axis subtracts; the integral is signed area, not total geometric area, unless you account for it.
An Honest Limitation
The method needs an antiderivative that can be found in elementary terms. Many innocent-looking functions have none, in which case numerical approximation is the honest path. This tool gives study estimates, not certified engineering results.