## Arc Length and Surface Area

Infinitesimal calculus provides us general formulas for the arc length of a curve and the surface area of a solid.

### Learning Objectives

Use integration to find the surface area of a solid rotated around an axis and the surface area of a solid rotated around an axis

### Key Takeaways

#### Key Points

• For a curve represented by $f(x)$ in range $[a,b]$, arc length $s$ is give as $s = \int_{a}^{b} \sqrt { 1 + [f'(x)]^2 }\, dx$.
• If a curve is defined parametrically by $x = X(t)$ and $y = Y(t)$, then its arc length between $t = a$ and $t = b$ is $s = \int_{a}^{b} \sqrt { [X'(t)]^2 + [Y'(t)]^2 }\, dt$.
• For rotations around the $x$– and $y$-axes, surface areas $A_x$ and $A_y$ are given, respectively, as the following: $A_x = \int 2\pi y \, ds, \,\, ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx \\ \\ A_y = \int 2\pi x \, ds, \,\, ds=\sqrt{1+\left(\frac{dx}{dy}\right)^2}dy$

#### Key Terms

• surface area: the total area on the surface of a three-dimensional figure
• curve: a simple figure containing no straight portions and no angles

Determining the length of an irregular arc segment is also called rectification of a curve. Historically, many methods have been used for specific curves. The advent of infinitesimal calculus led to a general formula, which we will learn in this atom. We will also use integration to calculate the surface area of a three-dimensional object.

### Arc Length

Consider a real function $f(x)$ such that $f(x)$ and $f'(x)=\frac{dy}{dx}$ (its derivative with respect to $x$) are continuous on $[a, b]$. The length $s$ of the part of the graph of $f$ between $x = a$ and $x = b$ can be found as follows.

Consider an infinitesimal part of the curve $ds$ (or consider this as a limit in which the change in $s$ approaches $ds$). According to Pythagoras’s theorem $ds^2=dx^2+dy^2$, from which:

$\displaystyle{\frac{ds^2}{dx^2}=1+\frac{dy^2}{dx^2} \\ ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx \\ s = \int_{a}^{b} \sqrt { 1 + [f'(x)]^2 }\, dx}$

Approximating Deltas: For a small piece of curve, $\Delta s$ can be approximated with the Pythagorean theorem.

If a curve is defined parametrically by $x = X(t)$ and y = Y(t), then its arc length between $t = a$ and $t = b$ is:

$\displaystyle{s = \int_{a}^{b} \sqrt { [X'(t)]^2 + [Y'(t)]^2 }\, dt}$

This is more clearly a consequence of the distance formula, where instead of a $\Delta x$ and $\Delta y$, we take the limit. A useful mnemonic is:

$\displaystyle{s = \int_{a}^{b} \sqrt { dx^2 + dy^2 } = \int_{a}^{b} \sqrt { \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 }\,dt}$

### Surface Area

For rotations around the $x$– and $y$-axes, surface areas $A_x$ and $A_y$ are given, respectively, as the following:

$\displaystyle{A_x = \int 2\pi y \, ds, \,\, ds=\sqrt{1+\left(\frac{dy}{dx}\right)^2}dx \\ \\ A_y = \int 2\pi x \, ds, \,\, ds=\sqrt{1+\left(\frac{dx}{dy}\right)^2}dy }$

### Example

For a circle $f(x) = \sqrt{1 -x^2}, 0 \leq x \leq 1$, calculate the arc length.

The curve can be represented parametrically as $x=\sin(t), y=\cos(t)$ for $0 \leq t \leq \frac{\pi}{2}$. Therefore:

$\displaystyle{s = \int_0^{\frac{\pi}{2}}\sqrt{\cos^2(t)+\sin^2(t)} = \frac{\pi}{2}}$

Now, calculate the surface area of the solid obtained by rotating $f(x)$ around the $x$-axis:

$\displaystyle{A_x = \int_{0}^{1} 2\pi \sqrt{1-x^2}\cdot \sqrt{1+\left(\frac{-x}{\sqrt{1-x^2}}\right)^2} \, dx = 2\pi}$

## Area of a Surface of Revolution

If the curve is described by the function $y = f(x) (a≤x≤b)$, the area $A_y$ is given by the integral $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.

### Learning Objectives

Use integration to find the area of a surface of revolution

### Key Takeaways

#### Key Points

• A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis.
• If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral $A_y = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt$.
• If the curve is described by the function $y = f(x), a \leq x \leq b$, then the integral becomes $A_x = 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$ for revolution around the $x$-axis.
• Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis.

#### Key Terms

• torus: the standard representation of such a space in three-dimensional Euclidean space; a shape consisting of a ring with a circular cross-section; the shape of an inner tube or hollow doughnut
• euclidean space: ordinary two- or three-dimensional space (and higher dimensional generalizations), characterized by an infinite extent along each dimension and a constant distance between any pair of parallel lines
• revolution: rotation: the turning of an object around an axis

A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane, known as the axis. Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is co-planar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis. A circle that is rotated about a diameter generates a sphere, and if the circle is rotated about a co-planar axis other than the diameter it generates a torus.

Surface of Revolution: A portion of the curve $x=2+\cos z$ rotated around the $z$-axis (vertical in the figure).

If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$ and the axis of revolution the $y$-axis, then the area $A_y$ is given by the integral:

$\displaystyle{A_y = 2 \pi \int_a^b x(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt}$

provided that $x(t)$ is never negative between the endpoints $a$ and $b$. The quantity $\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2$ comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. Likewise, when the axis of rotation is the $x$-axis, and provided that $y(t)$ is never negative, the area is given by:

$\displaystyle{A_x = 2 \pi \int_a^b y(t) \ \sqrt{\left({dx \over dt}\right)^2 + \left({dy \over dt}\right)^2} \, dt}$

If the curve is described by the function $y = f(x)$, $a \leq x \leq b$, then the integral becomes:

$A_x = 2\pi\int_a^b y \sqrt{1+\left(\frac{dy}{dx}\right)^2} \, dx \\ \quad= 2\pi\int_a^bf(x)\sqrt{1+\left(f'(x)\right)^2} \, dx$

for revolution around the $x$-axis, and

$A_y =2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} \, dy$

for revolution around the $y$-axis ($a \leq y \leq b$).

### Example

The spherical surface with a radius $r$ is generated by the curve $x(t) =r \sin(t)$, $y(t) = r \cos(t)$, when $t$ ranges over $[0,\pi]$. Its area is therefore:

\begin{align} A &{}= 2 \pi \int_0^\pi r\sin(t) \sqrt{\left(r\cos(t)\right)^2 + \left(r\sin(t)\right)^2} \, dt \\ &{}= 2 \pi r^2 \int_0^\pi \sin(t) \, dt \\ &{}= 4\pi r^2 \end{align}

## Physics and Engineering: Fluid Pressure and Force

Pressure is given as $p = \frac{F}{A}$ or $p = \frac{dF_n}{dA}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.

### Learning Objectives

Apply the ideas of integration to pressure

### Key Takeaways

#### Key Points

• The pressure is the scalar proportionality constant that relates the two normal vectors $d\mathbf{F}_n=-p\,d\mathbf{A} = -p\,\mathbf{n}\,dA$.
• For fluids near the surface of the earth, the formula may be written as $p = \rho g h$, where $p$ is the pressure, $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, and $h$ is the depth of the liquid in meters.
• Total force that the fluid pressure gives rise to is calculated as $\mathbf{F_n} = -(\int \rho g h \, dA) \, \mathbf{n}$.

#### Key Terms

• fluid: any substance which can flow with relative ease, tends to assume the shape of its container, and obeys Bernoulli’s principle; a liquid, gas, or plasma
• Gravitational acceleration: acceleration on an object caused by gravity; at different points on Earth, an acceleration between 9.78 and 9.82 m/s2, depending on altitude
• Pressure: the amount of force that is applied over a given area divided by the size of this area

Pressure ($p$) is force per unit area applied in a direction perpendicular to the surface of an object. While pressure may be measured in any unit of force divided by any unit of area, the SI unit of pressure (the newton per square meter) is called the pascal (Pa).

Fluid Pressure and Force: Pressure as exerted by particle collisions inside a closed container.

Mathematically, $p = \frac{F}{A}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact.

Pressure is a scalar quantity. It relates the vector surface element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates the two normal vectors:

$d\mathbf{F}_n=-p\,d\mathbf{A} = -p\,\mathbf{n}\,dA$

The subtraction (–) sign comes from the fact that the force is considered towards the surface element while the normal vector points outward. The total force normal to the contact surface would be:

$\displaystyle{\mathbf{F}_n = \int d\mathbf{F}_n=- \int p\,d\mathbf{A} = - \int p\,\mathbf{n}\,dA}$

Pressure is an important quantity in the studies of fluid (for example, in weather forecast). For fluids near the surface of the earth, the formula may be written as $p = \rho g h$, where $p$ is the pressure, $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, and $h$ is the depth of the liquid in meters. Using this expression, we can calculate the total force that the fluid pressure gives rise to:

$\mathbf{F_n} = -(\int \rho g h \, dA) \, \mathbf{n}$

This equation, for example, can be used to calculate the total force on a submarine submerged in the sea.

## Physics and Engeineering: Center of Mass

For a continuous mass distribution, the position of center of mass is given as $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$.

### Learning Objectives

Apply the ideas of integration to the center of mass

### Key Takeaways

#### Key Points

• In physics, the center of mass (COM) of a distribution of mass in space is the unique point at which the weighted relative position of the distributed mass sums to zero.
• In the case of a system of particles Pi, i = 1, , n , each with mass, mi, which are located in space with coordinates ri, i = 1, , n, the coordinates R of the center of mass is $\mathbf{R} = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{r}_i$.
• If the mass distribution is continuous with respect to the density, ρ(r), within a volume, V, then it follows that $\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV$.

#### Key Terms

• centroid: the point at the center of any shape, sometimes called the center of area or the center of volume

### Center of Mass

In physics, the center of mass (COM) of a mass or object in space is the unique point at which the weighted relative position of the distributed mass sums to zero. In this case, the distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are simplified when formulated with respect to the COM.

### System of Particles

In the case of a system of particles $P_i, i = 1, \cdots, n$, each with a mass, $m_i$, which are located in space with coordinates $r_i, i = 1, \cdots, n$, the coordinates $\mathbf{R}$ of the center of mass satisfy the following condition:

$\displaystyle{\sum_{i=1}^n m_i(\mathbf{r}_i - \mathbf{R}) = 0}$

Solve this equation for $\mathbf{R}$ to obtain the formula

$\displaystyle{\mathbf{R} = \frac{1}{M} \sum_{i=1}^n m_i \mathbf{r}_i}$

where $M$ is the sum of the masses of all of the particles.

### Continuous Distribution

If the mass distribution is continuous with respect to the density, $\rho (r)$, within a volume, $V$, then the integral of the weighted position coordinates of the points in this volume relative to the center of mass, $\mathbf{R}$, is zero, that is:

$\displaystyle{\int_V \rho(\mathbf{r})(\mathbf{r}-\mathbf{R})dV = 0}$

Solve this equation for the coordinates $\mathbf{R}$ to obtain:

$\displaystyle{\mathbf R = \frac 1M \int_V\rho(\mathbf{r}) \mathbf{r} dV}$

where $M$ is the total mass in the volume. If a continuous mass distribution has uniform density, which means $\rho$ is constant, then the center of mass is the same as the centroid of the volume.

Two Bodies and the COM: Two bodies orbiting the COM located inside one body. COM can be defined for both discrete and continuous systems. The two objects are rotating around their center of mass.

## 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 Objectives

Apply the ideas behind integration to economics and biology

### Key Takeaways

#### 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.

#### Key Terms

• flux: the rate of transfer of energy (or another physical quantity) through a given surface, specifically electric flux, magnetic flux
• 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

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.

## Probability

Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.

### Learning Objectives

Apply the ideas of integration to probability functions used in statistics

### Key Takeaways

#### Key Points

• The probability of $X$ to be in a range $[a,b]$ is given as $P [a \leq X \leq b] = \int_a^b f(x) \, \mathrm{d}x$, where $f(x)$ is the probability density function in this case.
• The integral of the partial distribution function over the entire range of the variable is 1.
• The standard normal distribution has probability density $f(X;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{X-\mu}{\sigma}\right)^2 }$.

#### Key Terms

• probability density function: any function whose integral over a set gives the probability that a random variable has a value in that set

Integration is commonly used in statistical analysis, especially when a random variable takes a continuum value. Here, we will learn what probability distribution function is and how it functions with regard to integration.

In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability for the random variable to fall within a particular region is given by the integral of this variable’s probability density over the region. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

### Probability Density Function

A probability density function is most commonly associated with absolutely continuous univariate distributions.

For a continuous random variable $X$, the probability of $X$ to be in a range $[a,b]$ is given as:

$\displaystyle{P [a \leq X \leq b] = \int_a^b f(x) \, \mathrm{d}x}$

where $f(x)$ is the probability density function in this case.

The integral of the pdf in the range $[-\infty, \infty]$ is

$\displaystyle{\int_{-\infty}^{\infty} f(x) \, \mathrm{d}x \, = \, 1}$

The expected value of $X$ (if it exists) can be calculated as:

$\displaystyle{E[X] = \int_{-\infty}^\infty x\,f(x)\,dx}$

### Example: Normal Distribution

Probability Distribution Function: Probability distribution function of a normal (or Gaussian) distribution, where mean $\mu=0$  and variance $\sigma^2=1$.

The standard normal distribution has probability density

$\displaystyle{f(X;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{X-\mu}{\sigma}\right)^2 }}$

This probability distribution has the mean and variance, denoted by $\mu$ and $\sigma ^2$, respectively. As shown below, the probability to have $x$ in the range $[\mu - \sigma, \mu + \sigma]$ can be calculated from the integral

$\displaystyle{\frac{1}{\sigma\sqrt{2\pi}} \int_{\mu-\sigma}^{\mu+\sigma} e^{ -\frac{1}{2}\left(\frac{X-\mu}{\sigma}\right)^2 } \approx 0.682}$

## Taylor Polynomials

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of the function’s derivatives.

### Learning Objectives

Use the Taylor series to approximate an integral

### Key Takeaways

#### Key Points

• The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number a is the power series $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n}$.
• Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.
• Taylor series can be used to evaluate an integral when there is no other integration technique available (other than numerical integration).

#### Key Terms

• series: the sum of the terms of a sequence
• polynomial: an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power

Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. The Taylor series of a real or complex-valued function $f(x)$ that is infinitely differentiable in a neighborhood of a real or complex number $a$ is the power series

$\displaystyle{f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(a)}{n! } \, (x-a)^{n}}$

where $n!$ denotes the factorial of $n$ and $f^{(n)}(a)$ denotes the $n$th derivative of $f$ evaluated at the point $x=a$. Any finite number of initial terms of the Taylor series of a function is called a Taylor polynomial.

Exponential Function as a Taylor Series: The exponential function (in blue) and the sum of the first 9 terms of its Taylor series at 0 (in red).

### Example

The Taylor series for the exponential function $e^x$ at $a=0$ is:

$\displaystyle{e^x = \sum_{n=0}^{\infty} \frac{x^n}{n! } = 1 + \frac{x^1}{1! } + \frac{x^2}{2! } + \frac{x^3}{3! } + \cdots}$

### Using Taylor Series to Evaluate an Integral

Taylor series can be used to evaluate an integral when there is no other integration technique available (of course, other than numerical integration). Let’s assume that the integration of a function ($f(x)$) cannot be performed analytically. To evaluate the integral $I = \int_{a}^{b} f(x) \, dx$, we can Taylor-expand $f(x)$ and perform integration on individual terms of the series. Since $f(x) = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n! } \, x^{n}$, we get:

$\displaystyle{I = \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{n! } \, \int_{a}^{b}x^{n}\, dx \\ \, \,= \sum_{n=0} ^ {\infty} \frac {f^{(n)}(0)}{(n+1)! } \, (b^{n+1}-a^{n+1})}$

Therefore, as long as Taylor expansion is possible and the infinite sum converges, the definite integral ($I$) can be evaluated.