Applications to Economics and Biology

Calculus has broad applications in diverse fields of science; examples of integration can be found in economics and biology.

Learning Objective

Apply the ideas behind integration to economics and biology

Key Points

Consumer surplus is thus the definite integral of the demand function with respect to price, from the market price to the maximum reservation price $CS = \int^{P_{\mathit{max}}}_{P_{\mathit{mkt}}} D(P)\, dP$.
The total flux of blood through a vessel with a radius $R$ can be expressed as $F = \int_{0}^{R} 2\pi r \, v(r) \, dr$, where $v(r)$ is the velocity of blood at $r$.
Calculus, in general, has broad applications in diverse fields of science.

Terms

cardiovascular
Relating to the circulatory system, that is the heart and blood vessels.
surplus
specifically, an amount in the public treasury at any time greater than is required for the ordinary purposes of the government
flux
the rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux, magnetic flux

Full Text

Calculus, in general, has a broad applications in diverse fields of science, finance, and business. In this atom, we will see some examples of applications of integration in economics and biology.

Consumer Surplus

In mainstream economics, economic surplus (also known as total welfare or Marshallian surplus) refers to two related quantities. Consumer surplus is the monetary gain obtained by consumers; they are able to buy something for less than they had planned on spending. Producer surplus is the amount that producers benefit from selling at a market price that is higher than their lowest price, thereby making more profit .

Supply and Demand Chart

Graph illustrating consumer (red) and producer (blue) surpluses on a supply and demand chart.

In calculus terms, consumer surplus is the derivative of the definite integral of the demand function with respect to price, from the market price to the maximum reservation price—i.e. the price-intercept of the demand function:

$\displaystyle{CS = \int^{P_{\mathit{max}}}_{P_{\mathit{mkt}}} D(P)\, dP}$

where $D(P)$ is a demand curve as a function of price.

Blood Flow

The human body is made up of several processes, all carrying out various functions, one of which is the continuous running of blood in the cardiovascular system. If we wanted, we could obtain a general expression for the volume of blood across a cross section per unit time (a quantity called flux). Since we can assume that there is a cylindrical symmetry in the blood vessel, we first consider the volume of blood passing through a ring with inner radius $r$ and outer radius $r+dr$ per unit time ($dF$):

$dF = (2\pi r \, dr)\, v(r)$

where $v(r)$ is the speed of blood at radius $r$. Here, $2 \pi r \,dr$ is the area of the ring. Therefore, the total flux $F$ is written as:

$\displaystyle{F = \int_{0}^{R} 2\pi r \, v(r) \, dr}$

where $R$ is the radius of the blood vessel. Once we have an (approximate) expression for $v(r)$, we can calculate the flux from the integral.

Blood Flow

(a) A tube; (b) The blood flow close to the edge of the tube is slower than that near the center.

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