Equation field guide
Integrals: accumulated change and area
An integral adds many small contributions. A definite integral accumulates a quantity over an interval; geometrically it can represent signed area under a curve.
The parts of an integral
In ∫ₐᵇ f(x) dx, f(x) is the quantity being accumulated, x is the variable, and a and b are the lower and upper bounds. dx signals that the accumulation is over tiny changes in x.
A definite integral produces one number. An indefinite integral, written ∫f(x) dx, describes a family of antiderivatives and includes a constant C.
Accumulation in context
If f(t) is velocity, integrating it over time gives displacement. If f(x) is a density, integrating it over distance gives a total amount. The units multiply: velocity times time gives distance.
Areas below the horizontal axis count as negative in an ordinary integral. For total geometric area, a problem may instead require absolute values or separate regions.
Its connection to derivatives
Differentiation measures local change; integration accumulates it. Under suitable conditions, the Fundamental Theorem of Calculus links them: ∫ₐᵇ f(x) dx = F(b) − F(a) when F′ = f.
This connection lets us turn a rate into a total and, conversely, recover a rate from an accumulated quantity.