An integral adds up something that is changing: water flowing into a tank, distance from a speedometer, area under a curve. Here is the idea, then the theorem that makes it easy to compute.
Adding up a rate
Suppose water flows into a tank at a rate that changes over time. If it were constant, total water = rate × time. When the rate changes, chop time into short pieces, treat the rate as constant on each, and add the pieces. Each piece is a thin rectangle under the rate curve, so the total is the area under the curve. Making the pieces infinitely thin gives the exact answer: the definite integral.
Now let the right-hand end move. Call the area from 0 up to x the accumulation functionA(x). Drag x and watch both graphs:
Rate: f(t) (area up to x shaded)
Amount so far: A(x)=∫0xf(t)dt
Area so far A(x) = 4.2017 · slope of A at x = 1.4977 = height of f at x
The left graph is a rate (like litres per minute). The right graph adds it up. Wherever f dips below zero, the amount starts to fall. The slope of the right curve is always the height of the left one: that is the Fundamental Theorem of Calculus.
The Fundamental Theorem of Calculus
Look at the ruler on the right graph. Its slope always equals the height of the left curve at the same x. That is not a coincidence: push x a tiny bit further and the extra area is a thin strip of height f(x), so area grows at rate f(x). In symbols, A′(x)=f(x).
So the area function is an antiderivative of f, which means you can compute areas without adding up any rectangles at all:
∫abf(x)dx=F(b)−F(a)where F′=f
Finding F is the skill of integration. The main techniques are on integration rules, and the integral calculator works any integral step by step.
Negative area
In the explainer, where f dips below zero the accumulated amount goes down: water draining out. Integrals count area below the axis as negative. When you want the total unsigned area, split at the zeros; the area under the curve calculator does this for you.
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Questions students ask
What is an integral in simple terms?
An integral adds up a quantity that changes continuously. On a graph, the definite integral ∫abf(x)dx is the signed area between the curve and the x-axis from a to b.
What is the difference between definite and indefinite integrals?
A definite integral has bounds and gives a number (an area or total). An indefinite integral has no bounds and gives a family of functions, the antiderivatives F(x)+C.
What does the ∫ symbol mean?
It is an elongated S, for “sum”, introduced by Leibniz. The dx stands for the width of the infinitely thin slices being added up.
What is the Fundamental Theorem of Calculus?
It links the two halves of calculus: the rate of change of the accumulated area A(x)=∫axf(t)dt is f(x) itself, and so ∫abf(x)dx=F(b)−F(a) for any antiderivative F.
What are integrals used for?
Distance from velocity, total water from a flow rate, work from a force, probability from a density, volumes of solids and average values. Anything where a rate is added up over time or space.