Definite integral — the area under the graph with the Newton–Leibniz formula
Function f(x)
Value of the integral
The definite integral ∫ f(x) dx from a to b is the area between the graph and the X axis over the interval [a; b]. Parts below the axis are counted with a minus sign.
The Newton–Leibniz formula
The main formula of integral calculus links the definite integral with the antiderivative: ∫ f(x) dx from a to b = F(b) − F(a), where F is any antiderivative of f. So the area can be found without sums and limits: just take the antiderivative and substitute the bounds.
📚 Theory: what an integral is
The definite integral ∫ₐᵇ f(x) dx is the area of the figure between the graph of f(x) and the X axis over the interval from a to b. Imagine the figure sliced into thousands of narrow vertical strips: the area of each ≈ f(x)·(width), and the integral is the sum of all strips as the slicing becomes infinitely thin.
Physical meaning: if f(x) is velocity, the integral is the total distance travelled; if f(x) is power, the integral is energy. The integral accumulates a quantity whose rate is given by the integrand.
The sign of the area: parts of the graph below the X axis contribute negatively. That is why, for example, ∫ sin x dx over a full period equals zero: the area above cancels the area below.
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