Business Applications of Integration

Because integration reverses differentiation, and because marginal cost and marginal revenue are THEMSELVES derivatives (of total cost and total revenue respectively — the subject of the earlier Differentiation chapter), integration is precisely the tool that recovers a TOTAL function from a MARGINAL one. This closing section of the CHSE Odisha Class 11 Business Mathematics and Statistics chapter on Integration applies every technique covered above to genuine commerce problems.

Total cost from marginal cost

If $MC(x)$ is the marginal cost function (the rate of change of total cost with respect to output $x$), then the total cost function is

The constant of integration is fixed using the fact that total cost at zero output equals the fixed cost: $TC(0) = \text{Fixed Cost}$.

Total revenue from marginal revenue

If $MR(x)$ is the marginal revenue function, then the total revenue function is

Here the constant of integration is fixed differently — total revenue at zero output is always zero (no units sold, no revenue earned), so $TR(0)=0$. Once $TR(x)$ is known, the demand (average revenue) function is $p(x) = \dfrac{TR(x)}{x}$.

Area under a curve

For a function $f(x) \ge 0$ on $[a,b]$, the definite integral $\displaystyle\int_{a}^{b} f(x)\,dx$ gives exactly the area bounded by the curve $y=f(x)$, the $x$-axis, and the two vertical lines $x=a$ and $x=b$ — this is the geometric meaning behind the definite integral, and it is what makes consumer's and producer's surplus (below) representable as areas on a demand-supply diagram.

<!-- FIGURE-NEEDED: graph — a shaded region under a curve y=f(x) between x=a and x=b, bounded below by the x-axis and on the sides by the two vertical lines x=a and x=b, illustrating that the definite integral equals this area -->

Consumer's surplus and producer's surplus

At the market equilibrium point $(x_{0}, p_{0})$ — where the demand curve $p=D(x)$ and the supply curve $p=S(x)$ intersect — buyers who would have been willing to pay MORE than $p_{0}$ gain a benefit, and sellers who would have been willing to accept LESS than $p_{0}$ gain a benefit too. Integration measures both:

$CS$ is the area between the demand curve and the horizontal line $p=p_{0}$, from $x=0$ to $x=x_{0}$; $PS$ is the area between that same horizontal line and the supply curve, over the same interval. …