What an integral means: area, accumulation and the fundamental theorem

A definite integral adds up a quantity that changes continuously. Geometrically, ∫ₐᵇ f(x) dx is the area between the graph of f and the x-axis from x = a to x = b, with area below the axis counted as negative. Physically, it turns a rate into a total: integrate speed over time and you get distance. The integral calculator works out definite integrals numerically, and this guide explains what the number means.

Area under a curve

Areas of rectangles and triangles are easy. The area under a curve is not, because the top edge keeps changing height. Take y = x² from x = 0 to x = 3. The region starts flat at 0 and rises to a height of 9. Its area is exactly 9 square units, but no school formula for shapes gives you that. The integral is the tool that does.

Adding up thin strips

The idea behind the integral sign, a stretched S for sum, is to slice the region into thin vertical strips, treat each as a rectangle, and add them up. For x² from 0 to 3:

• with 6 strips, using each strip's left edge for the height, the total is 6.875; using the right edge, 11.375;
• with 30 strips, the two totals are 8.555 and 9.455;
• with 300 strips, 8.955 and 9.045.

As the strips get thinner, both totals squeeze towards 9. These are called Riemann sums, and the integral is defined as their limit. The dx in the notation is a reminder of the strip width. The summation calculator can add up sums like these if you want to watch them converge.

The fundamental theorem

Adding infinitely many strips sounds impossible, but the fundamental theorem of calculus provides a shortcut. If F is a function whose derivative is f, then

∫ₐᵇ f(x) dx = F(b) − F(a).

Integration is the reverse of differentiation. The derivative of x³/3 is x², so the area under x² from 0 to 3 is 3³/3 − 0³/3 = 27/3 = 9, matching the strips. The derivative of −cos x is sin x, so the area under one arch of the sine curve, from 0 to π, is −cos π − (−cos 0) = 1 + 1 = 2. The rules for derivatives therefore become rules for integrals run backwards; How to find a derivative: the rules, with worked examples covers them, and the derivative calculator lets you check an antiderivative by differentiating it.

Signed area and accumulation

Where the curve dips below the x-axis, the integral counts that area as negative. The integral of sin x from 0 to 2π is 0, not 4: the arch above the axis and the arch below cancel. That sounds like a nuisance, but it is exactly what accumulation needs. If f is a velocity, positive means moving forward and negative means moving back, so the integral gives the net change in position. Walk forward at 3 m/s for 10 seconds and the integral of velocity is 30 m. If you want the total area regardless of sign, integrate the absolute value, |f(x)|, instead.

The same pattern turns flow rate into volume, power into energy and a probability density into a probability.

When there is no formula

Many functions have no antiderivative that can be written with ordinary functions. The bell-curve shape e^(−x²) is the famous example, yet its integral from 0 to 1 is a perfectly definite number, about 0.746824. This is why the calculator works numerically rather than symbolically. It uses adaptive Gauss–Kronrod quadrature: instead of equal strips, it samples the function at carefully chosen points, compares two estimates to judge its own error, and splits the interval wherever the function is hard to follow. For smooth functions it typically reaches 10 to 12 significant digits and shows an error estimate and a shaded plot. It needs finite limits, which can be expressions like pi/2. Functions that blow up inside the interval, or have jumps, can give misleading results, so plot them first with the graphing calculator.

Definite and indefinite integrals

A definite integral, with limits, is a number. An indefinite integral, without limits, is a family of functions: ∫ x² dx = x³/3 + C. The + C is there because every constant has derivative 0, so x³/3 + 5 is just as valid an antiderivative as x³/3. When you evaluate F(b) − F(a), the constant cancels, which is why definite integrals do not need it.