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How to calculate the integral

1. Enter the function f(x). For example x^2 or sin(x).
2. Enter the limits. The lower and upper limits define the interval over which the area is calculated.
3. Calculate. You get the value of the definite integral, i.e. the area between the curve and the x-axis.

What does the definite integral mean?

The definite integral ∫ₐᵇ f(x) dx describes the signed area between the function's curve and the x-axis over the interval [a, b]. When the curve is above the x-axis, the area is positive; below the axis the area is counted as negative. The integral is a central tool in calculating areas, volumes, work and many physical quantities.

Simpsonin menetelmä

The calculator uses Simpson's rule, which estimates the area by fitting parabolic arcs to the curve. The method is very accurate for smooth functions and gives a practically correct result for ordinary problems. If the function has a discontinuity or a vertical asymptote in the chosen interval, the result may be unreliable — in that case divide the interval into parts.

Example: distance from velocity

If velocity as a function of time is v(t) = 2t (m/s), the distance travelled over t = 0…5 is the definite integral of velocity ∫₀⁵ 2t dt. Geometrically this is the area of the triangle under the velocity graph: base 5 and height v(5) = 10, so the area = ½ · 5 · 10 = 25 metres. The calculator gives the same result numerically. Thus the integral connects velocity and distance in the same way that the derivative connects distance and velocity — they are inverse operations of each other.

Definite and indefinite integral

The indefinite integral ∫ f(x) dx means a function whose derivative is f(x); it always involves a constant term + C. The definite integral ∫ₐᵇ f(x) dx is a number obtained by substituting the limits and subtracting. The fundamental theorem of calculus ties these together: the definite integral is the difference of the antiderivative's values at the limits. This calculator focuses on the definite integral, i.e. the area over a given interval, which is enough for most practical and school problems.

Pinta-alasta tilavuuteen ja keskiarvoon

The definite integral calculates the area under the curve, but much else follows from the same idea. When the curve is rotated around an axis, the integral gives the volume of the resulting solid — this is how the volumes of solids of revolution are calculated, for example. In physics the integral turns velocity into distance and force into work done, because these are "accumulations" over time or distance.

The integral is also used to calculate the average of a function over an interval: it is the integral divided by the length of the interval. This gives, for example, the average of a varying temperature or power over a certain period. In all of these the same idea of continuous summation recurs: the integral adds up infinitely many small parts. This calculator gives the area numerically, and once you recognise what the area represents in each situation, the same tool serves many kinds of problems.

Good to know about the integral

• A definite integral is a number, an indefinite integral is a function (+ constant C).
• If the upper and lower limits are the same, the value of the integral is zero.
• Swapping the limits changes the sign of the integral.
• The part below the x-axis is counted as negative area.

संक्षेप में

The integral calculator computes the definite integral, i.e. the area under the curve, with the given limits. Enter the function and the limits, and the tool computes the result accurately with Simpson's rule. The same idea of area generalises to volumes, work and averages, so the calculator serves many kinds of problems.

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