Unit 3: Applications

Preface

This is the note of the MIT Course 18.01.2 Calculus. For more details, refer to MIT 18.x Catalog, and it can also be found at MIT Open Learning Library.

Area and Volumes

Process for Finding Areas and Volumes

  1. Draw a picture
  2. Identify limits of integration
  3. Identify integrand
  4. Integrate to solve (as needed)

Volumes by Slicing

The volume of a slive of bread is $$ dV=A(x)\ dx $$ where $dx$ is the width of the slice, and $A(x)$ is the cross-sectional area. The volume of the entire loaf is $$ V=\int A(x)\ dx $$

Method of Disks

The method of disks gives the differential volume element for a curve rotated about the $x$-axis $$ dV=\pi y^2\ dx $$

Method of Shells

The method of shells for a curve $y=f(x)$ rotated about the $y$-axis $$ dV=2\pi x y \ dx $$

Method of Washers

The volume of an infinitesimal washer is given by $$ dV=\pi (y_{outer}^2-y_{inner}^2)\ dx $$

Average Value

Average Value of $f(x)$

The average value of a function $f(x)$ on an interval $a\le x\le b$ is a given by $$ \frac{1}{b-a}\int_a^b f(x)\ dx $$

Average Value And Average Rate of Change

Fundamental Theorem of calculus tells us that the average rate of change of $y$ over $a\le t\le b$ is the same as the average value of $a\le t\le b$ $$ \text{ave}(y^{\prime})=\frac 1 {b-a}\int_a^b y^{\prime}(t)\ dt=\frac{y(b)-y(a)}{b-a}=\text{average rate of change of $y(x)$} $$

Weighted Average

The weighted average value of a function $f(x)$, weighted by a weight function $w(x)$ on an interval $a\le x\le b$ is given by $$ \frac{\int_a^bf(x)w(x)\ dx}{\int_a^bw(x)\ dx} $$

Center of Mass

1-Dimensional

For a 1-dimensional rod of length $L$ with linear density function $\lambda(x)$ (measured in mass per unit length), the center of mass is the weighted average: $$ x_c=\frac{\int_0^Lx\lambda(x)\ dx}{\int_0^L\lambda(x)\ dx} $$

2-Dimensional

For a 2-dimensional region whose mass is proportional to area, the center of mass, or centroid $(\overline{x},\overline{y})$, is the weighted average of the position with respect to the area. The $x-$ and $y-$ coordinates of the centroid are computed by the following integrals: $$ \begin{aligned} \overline{x}=&\frac{\int_a^bx\ dA}{\int_a^b\ dA}\\ \overline{y}=&\frac{\int_a^by\ dA}{\int_a^b\ dA} \end{aligned} $$ where $dA$ is the infinitesimal area, and we must define the limits of integration so that we integrate over the entire region with respect to $x$ or $y$ respectively.

Work, Heat, And Energy

Work is done on an object when energy is transferred to that object.

Work, heat, and energy have the same units, but we typically measure them differently.

  • We measure work in Newton-meters (N⋅M)
  • We measure heat in calories (cal)
  • We measure energy in Joules (J)

James Joule recognized and measured the relationship between heat energy and mechanical work. $$ 1\text{ cal}=4.186\text{ J}=4.186\text{ N$\cdot$m} $$ The approach to solving problems involving work, heat, or energy is the same, and involves computing an integral.

A common example of mechanical work is when a force is applied over a distance. This force applied to the object transfers energy in the form of kinetic or potential energy. In this case, the formula for work is given by $$ \text{W}=\int F\ dx $$ In the cauldron example, heat energy was transferred to the water in the cauldron. We measured the work done, or the energy transferred, by measuring the total change in tempertaure $T$: $$ \int T\ dV $$

Probability

Discrete Probability

$$ \text{Discrete Probability}=\frac{\text{# specified outcomes}}{\text{# all possible outcomes}} $$

Note this means that the probability is a positive fraction that is less than or equal to $1$.

Distributions

A distribution is a nonnegative function $w(x)$ that approximates the discrete distribution. The area under a distribution measures the probability that an event occurs. $$ P(a<x<b)=\frac{\int_a^bw(x)\ dx}{\int_{-\infty}^{+\infty}w(x)\ dx} $$ Since $\displaystyle\int_{-\infty}^{+\infty}w(x)\ dx=C$ is a finite constant, to scale the function, we define $\displaystyle p(x)=\frac{w(x)}{C}$, which had the property that $\displaystyle\int_{-\infty}^{+\infty}p(x)\ dx=1$

A probability distribution or a probability density function is defined as a nonnegative function $p(x)$ defined on $-\infty <x<\infty$ with the property that $\displaystyle\int_{-\infty}^{+\infty}p(x)\ dx=1$. In this case, the probability of the outcome $a<x<b$ is defined as: $$ P(a<x<b)=\int_a^bp(x)\ dx $$

Normal Distribution

$$ p(x)=\frac1{\sigma \sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}},\qquad -\infty<x<+\infty $$

where the expected value is $\mu$ and the standard deviation is $\sigma$.

Exponential Distribution

$$ p(t)= \begin{cases} 0,&\quad t<0 \\ a e^{-a t},&\quad 0\le t<\infty \end{cases} $$

where the expected value is $1/a$.

Expected Value

Given a probability distribution $p(x)$ such that $\displaystyle\int_{-\infty}^\infty p(x)\ dx=1$, the expected value, $E$, also known as the mean. $$ \text{Expected value}=E=\int_{-\infty}^\infty x p(x)\ dx $$ Note that the expected value is completely analogous to center of mass.

Variance

Given a probability distribution $p(x)$ with $\displaystyle\int_{-\infty}^{\infty}p(x)\ dx=1$, and whose expected value is $E$, the variance, $V$, measures the average spread of outcomes from the expected value. $$ \text{Variance}=V=\int_{-\infty}^\infty (x-E)^2p(x)\ dx $$ The standard deviation, $\sigma$, is the square root of the variance: $$ \sigma=\sqrt{V} $$

Area Under The Bell Curve

$$ \text{Area} = \int_0^\infty x^{-x^2}\ dx=\frac{\sqrt{\pi}}{2} $$

Numerical Integration

Riemann sums

To approximate the integral $\displaystyle\int_a^b f(x)\ dx$:

  • Subdivide the interval into $n$ pieces $a=x_0<x_1<\cdots<x_{n-1}<x_n=b$.
  • Let $y_0=f(x_0)$, $y_1=f(x_1)$, … , $y_{n-1}=f(x_{n-1})$, and $y_n=f(x_n)$.
  • $\displaystyle \Delta x=\frac{b-a}{n}$.

Left Riemann Sum

$$ \int_a^b f(x)\ dx\approx \Delta x(y_0+y_1+\cdots+y_{n-1}) $$

Right Riemann Sum

$$ \int_a^b f(x)\ dx\approx \Delta x(y_1+y_2+\cdots+y_{n}) $$

Midpoint Rule

Let $$ \tilde{y}_1=f\left(\frac{x_0+x_1}2\right),\quad \tilde{y}_1=f\left(\frac{x_1+x_2}2\right),\ \dots\ ,\quad \tilde{y}_n=f\left(\frac{x_{n-1}+x_n}2\right) $$ then $$ \int_a^bf(x)\ dx\approx \Delta x\left(\tilde{y}_1+\tilde{y}_2+\cdots+\tilde{y}_n\right) $$

Trapezoidal Rule

$$ \int_a^bf(x)\ dx\approx \Delta x\left(\frac 1 2y_0+y_1+\cdots+y_{n-1}+\frac 1 2 y_n\right) $$

Simpson’s Rule

$$ \int_a^bf(x)\ dx\approx \frac {\Delta x}3 \left(y_0+4y_1+2y_2+4y_3+\cdots+2y_{n-2}+4y_{n-1}+y_n\right) $$

Error Bounds

Error bound of Trapezoidal Rule

Suppose $\vert f^{\prime\prime}(x)\vert\le M$ for $a\le x\le b$. Then the error of the Trapezoidal Rule over $n$ subintervals is given by $$ \vert E_T\vert \le\frac{M(b-a)^3}{12n^2} $$

Error bound of Midpoint Rule

Suppose $\vert f^{\prime\prime}(x)\vert\le M$ for $a\le x\le b$. Then the error of the Midpoint Rule over $n$ subintervals is given by $$ \vert E_M\vert \le\frac{M(b-a)^3}{24n^2} $$

Error bound of Simpson’s Rule

Suppose $\vert f^{(4)}(x)\vert\le M$ for $a\le x\le b$. Then the error of the Simpson’s Rule over $n$ subintervals is given by $$ \vert E_S\vert\le\frac{M(b-a)^5}{180n^4} $$