Definite integrals and the calculation of areas

Definition of the definite integral

If f is continuous and non-negative on [a;b], the integral from a to b of f(x) dx represents the area, expressed in units of area, of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b. It is denoted as: integral from a to b of f(x) dx.

Calculation using an antiderivative

Fundamental Theorem of Calculus: if F is an antiderivative of f on [a;b], then

∫_a^b f(x) dx = F(b) - F(a)

Example: Calculate ∫_0^2 x² dx. An antiderivative of x² is F(x) = x³/3. Therefore, the integral is F(2) – F(0) = 8/3 – 0 = 8/3 units of area.

Sign of the integral

If f is negative on [a;b], the integral is negative: it then represents the opposite of the geometric area.

Example: the integral of -1 to 0 of x dx = [x²/2] from -1 to 0 = 0 – 1/2 = -1/2.

Common pitfall

Do not confuse the integral with the area when f changes sign over the interval. If f is negative on a part of [a;b], you must divide the interval at the points where f is zero, calculate each part, then take the absolute value of the negative parts to obtain the total geometric area.