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Business Mathematics and Statistics · Ch 2 — Integral Calculus – I (Indefinite/Definite Integrals)

Definite Integrals — the Fundamental Theorem and Key Properties

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Definite Integrals — the Fundamental Theorem and Key Properties

Evaluating a definite integral

The Fundamental Theorem of Calculus connects the indefinite integral (antiderivative) to a NUMBER — the definite integral over an interval [a,b][a,b]:

∫abf(x) dx=F(b)−F(a),where F′(x)=f(x)\int_a^b f(x)\,dx=F(b)-F(a),\quad\text{where } F'(x)=f(x)

Unlike an indefinite integral, a definite integral carries no arbitrary constant CC — it cancels out automatically in the subtraction F(b)−F(a)F(b)-F(a).

Key properties

∫abf(x) dx=−∫baf(x) dx∫abf(x) dx=∫acf(x) dx+∫cbf(x) dx (a<c<b)\int_a^b f(x)\,dx=-\int_b^a f(x)\,dx\qquad \int_a^b f(x)\,dx=\int_a^c f(x)\,dx+\int_c^b f(x)\,dx\ (a<c<b)

∫abf(x) dx=∫abf(a+b−x) dx∫0af(x) dx=∫0af(a−x) dx\int_a^b f(x)\,dx=\int_a^b f(a+b-x)\,dx\qquad \int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx

Note

A definite integral over [a,b][a,b] of a non-negative function equals the AREA under the curve

This geometric reading — the definite integral as the area between y=f(x)y=f(x), the xx-axis, and the vertical lines x=a,x=bx=a,x=b — is exactly what the next chapter builds on for genuine business applications: total cost from marginal cost, and consumer/producer surplus. …

Definition 1Definite Integral (Fundamental Theorem)

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx=F(b)-F(a), where FF is any antiderivative of ff; the arbitrary constant CC cancels …

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