The Definite Integral and the Fundamental Theorem

In the previous chapter, integration reversed differentiation: given a function $f$, its indefinite integral $\int f(x)\,dx = F(x) + c$ was itself a function (an antiderivative, plus an arbitrary constant $c$). The definite integral takes this one step further by evaluating an antiderivative between two fixed values of $x$, producing a single number rather than a function.

Notation. The definite integral of $f$ from $x = a$ to $x = b$ is written

where $a$ is the lower limit and $b$ the upper limit of integration, and $f(x)$ is the integrand. Note carefully that a definite integral has no arbitrary constant $c$ — it evaluates to a definite numerical value.

The Fundamental Theorem of Calculus. This theorem is the bridge between antiderivatives and definite integrals. If $f$ is continuous on the closed interval $[a, b]$ and $F$ is any antiderivative of $f$ (that is, $F'(x) = f(x)$), then

The usual shorthand for $F(b) - F(a)$ is the square-bracket notation $\big[\,F(x)\,\big]_{a}^{b}$, read as "$F$ of $x$, evaluated from $a$ to $b$". So the working template is always: find an antiderivative, substitute the upper limit, subtract the value at the lower limit.

Why the constant $c$ can be dropped. If instead of $F(x)$ one used $F(x) + c$, the theorem would give $\big(F(b)+c\big) - \big(F(a)+c\big) = F(b) - F(a)$ — the constant cancels. This is why the arbitrary constant is simply omitted when evaluating a definite integral; any one antiderivative gives the same answer.

Meaning. Geometrically, $\int_a^b f(x)\,dx$ is the net signed area between the curve $y = f(x)$ and the $x$-axis from $x = a$ to $x = b$ — area above the axis counting as positive and area below as negative. This interpretation underlies the next chapter, Applications of Definite Integration.

Maharashtra's Std XII commerce Mathematics and Statistics syllabus draws on the same definite-integration principles — the Fundamental Theorem of Calculus, the standard integrals, and the properties below — that underpin integral calculus across Indian higher-secondary mathematics-and-statistics curricula, scoped here to polynomial, rational, exponential and logarithmic integrands rather than a full analysis course.