Integration

ECON 441: Introduction to Mathematical Economics
Lecture 13

Div Bhagia

Inverse of Differentiation

Path of population over time: \[P(t) = 2 t^{0.5}\]

Rate of change of population: \[P'(t)= \frac{d P}{dt} = t^{-0.5}\]

But what if instead we were given \(P'(t)\) and were tasked with finding \(P(t)\)?

Inverse of Differentiation

Note that \[P'(t)= \frac{d P}{dt} = t^{-0.5}\] is the derivative of \(P(t) = 2 t^{0.5}\), but also of \(P(t) = 2 t^{0.5} + 30\).

Generally, at best, we can find the following from just \(P'(t)\): \[P(t) = 2 t^{0.5} + c\]

However, if we have an initial condition such as \(P(0) = 50\), we can also find \(c\).

Example: From Marginal Cost to Total Cost

A firm’s marginal cost is \[MC(Q) = 6Q + 2\] and its fixed cost is 8, so \(C(0) = 8\).

Marginal cost is the derivative of total cost: \(MC(Q) = C'(Q)\).

Can we recover the total cost function \(C(Q)\)?

Integration

  • Integration is the reverse of differentiation

  • If \(f(x)\) is the derivative of \(F(x)\), we can integrate \(f(x)\) to find \(F(x)\): \[\frac{d}{d x} F(x)=f(x) \; \Rightarrow \; \int f(x)\, d x=F(x)+c\]

  • Rules of integration follow from rules of differentiation

Rules of Integration

Power rule \[\int x^n\, d x=\frac{1}{n+1} \cdot x^{n +1} + c \quad (n \neq-1)\]

Examples: \[\int x^3\, d x, \qquad \int x\, d x, \qquad \int 1\, d x\]

Rules of Integration

Exponential rule \[\int e^x\, d x=e^x+c\]

Log rule \[\int \frac{1}{x}\, d x=\ln x+c \quad (x>0)\]

Rules of Integration

Integral of a sum \[\int[f(x)+g(x)]\, d x=\int f(x)\, d x+\int g(x)\, d x\]

Integral of a multiple \[\int k f(x)\, d x=k \int f(x)\, d x\]

Example: \[\int (x^2 + 3x + 1)\, dx = \hspace{7em}\]

Definite Integrals

Definite integral: \[\int_a^b f(x)\, d x=\Big[F(x)\Big]_{a}^b=F(b)-F(a)\]

Example: \[\int_1^3 2 x^2\, dx = \hspace{7em}\]

Example: Investment and the Capital Stock

A firm’s capital stock \(K(t)\) grows through investment: \(I(t) = \dfrac{dK}{dt}\).

Suppose \(I(t) = 2t + 5\) and the initial capital stock is \(K(0) = 100\).

  • Find the path of the capital stock \(K(t)\).

  • How much capital is accumulated between \(t = 2\) and \(t = 4\)?

Area Under the Curve

A concave curve f of x over the interval from x zero to x five, with five grey rectangles of equal width standing under it, each as tall as the curve at its left edge; the fourth rectangle's width is marked delta x three and its height f of x three

\[\text{Area} \approx \sum_{i=1}^n f(x_i)\, \Delta x_i\]

Area Under the Curve

The same concave curve with twenty narrow grey rectangles under it, which together fill the region under the curve almost exactly

\[\begin{aligned} \text{Area} &= \lim_{n \rightarrow \infty} \sum_{i=1}^n f(x_i)\, \Delta x_i \\ &= \int_{x_1}^{x_n} f(x)\, dx \end{aligned}\]

Example: Consumer Surplus

Demand: \(p = 48 - Q^2\). Market price: \(p = 12\). Find the consumer surplus.

The demand curve p equals 48 minus Q squared, falling from a price of 48 at zero quantity and getting steeper as quantity rises, with dashed lines marking the market price of 12 on the price axis and the quantity of 6 on the quantity axis, meeting at a point on the curve

Homework Questions

  • Exercise 14.2: 1 (a), (c), (d)

  • Exercise 14.3: 1 (a), (e), 2 (a), (d), 5

Textbook reference: 14.1-14.3

Midterm 2 is next week: review on Monday, exam on Wednesday

A sample exam and a help sheet are posted on the Exams page