Business Mathematics and Statistics · Ch 3 — Integral Calculus – II (Area under curves; Application of Integration in Economics and Commerce)

Area Under a Curve Using Definite Integrals

Figure 1 — Area Under y = x² from x = 1 to x = 3

This Tamil Nadu HSC Class 12 Business Mathematics and Statistics chapter builds directly on the previous chapter's integration techniques, turning the definite integral into a genuinely practical tool: measuring the area under a curve, recovering total cost/revenue from marginal functions, and computing consumer's and producer's surplus — core commerce and economics applications.

The area formula

For a curve $y=f(x)$ that is non-negative on $[a,b]$ , the area enclosed between the curve, the $x$ -axis, and the vertical lines $x=a,x=b$ is

\text{Area}=\int_a^b f(x)\,dx

When finding the area BETWEEN two curves $y=f(x)$ (upper) and $y=g(x)$ (lower) over $[a,b]$ , subtract:

\text{Area}=\int_a^b\left[f(x)-g(x)\right]dx

Worked reasoning

For the area between $y=x$ and $y=x^2$ from $x=0$ to $x=1$ (where the line lies above the parabola on this interval, since $x\geq x^2$ for $0\leq x\leq1$ ):

\text{Area}=\int_0^1(x-x^2)\,dx=\left[\frac{x^2}{2}-\frac{x^3}{3}\right]_0^1=\frac12-\frac13=\frac16

Note

Always check WHICH curve is on top before subtracting

Subtracting the curves in the wrong order gives a NEGATIVE area — a definite integral computing an area between two curves must always be set up as (upper $-$ lower), confirmed by checking which function has the larger value somewhere in the interval.

Definition 1Area Under a Curve

For $f(x)\geq0$ on $[a,b]$ , the area between the curve $y=f(x)$ , the x-axis, and $x=a,x=b$ is $\int_a^b f(x)\,dx$ . Between two curves, it is $\int_a^b[f(x)-g(x)]\,dx$ with $f$ the upper curve.

← Previous chapterCh 2 — Integral Calculus – I (Indefinite/Definite Integrals)Next · 2 →Cost and Revenue Functions from Marginal Functions