Hyperbolic sine and cosine

\[ \begin{aligned} \sinh x &= \frac{e^x - e^{-x}}{2} \\ \cosh x &= \frac{e^x + e^{-x}}{2} \\ d\frac{\sinh x}{dx} &= \cosh x \\ d\frac{\cosh x}{dx} &= \sinh x \\ 1 + \sinh^2 x &= \cosh^2 x \\ \end{aligned} \]

Disc Area

3 ways to find the area of a disc:

\[ \begin{aligned} A &= \frac{1}{2}\int_0^{2\pi} r^2 \, d\theta \\ A &= \int_0^{R} 2\pi \, r \, dr \\ A &= 2\int_{-R}^{R} \sqrt{R^2 - x^2} dx \\ \end{aligned} \]

Polar area element

\[ \begin{aligned} dA &= \frac{1}{2}r^2\, d\theta = \frac{1}{2}(f(\theta))^2\, d\theta \end{aligned} \]

Volumes of Revolution

Use cylindrical shells when the area element is parallel to the axis of rotation:

\[ \begin{aligned} dV &= 2 \, \pi \, r \, h(r) \, dr \\ \end{aligned} \]

Use washer when the area element is perpendicular to the axis of rotation:

\[ \begin{aligned} dV &= \pi \, R(t)^2 - \pi \, r(t)^2\, dt \\ \end{aligned} \]

Arc Length

\[ \begin{aligned} L &= \int dL \\ dL &= \sqrt{1 + (\frac{dy}{dx})^2} \, dx \\ \end{aligned} \]

Parametric Curves

\[ \begin{aligned} dL &= \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt \end{aligned} \]

Surface area of revolution

\[ \begin{aligned} dS &= 2 \pi r \, dL \\ \end{aligned} \]

Revolving around the x-axis:

\[ \begin{aligned} dS = 2\pi \, f(x) \sqrt{1 + (\frac{dy}{dx})^2} \, dx \\ \end{aligned} \]

2 ways of revolving around the y-axis:

\[ \begin{aligned} dS &= 2\pi \, x \sqrt{1 + (\frac{dy}{dx})^2} \, dx \\ &= 2\pi \, f^{-1}(y) \sqrt{1 + (\frac{dx}{dy})^2} \, dy \\ \end{aligned} \]

Work

Work = Force x Distance.

Pull up a hanging rope

\[ \begin{aligned} dW &= \rho \, x \, dx \\ \end{aligned} \]

where \(x\) is measured from the top, and \(\rho\) is the weight density (or mass density multiplied by gravity.)

Spring

\[ \begin{aligned} dW &= F(x) \, dx \\ \end{aligned} \]

Elements

Mass of a rod

\[ \begin{aligned} dM &= \rho(x) \, dV = \rho(x) \, dx \\ \end{aligned} \]

Mass of a sphere

\[ \begin{aligned} dM &= \rho(r) \, dV = \rho(r) \, 4 \pi r^2 \, dr \\ \end{aligned} \]

Present Value

\[ \begin{aligned} P(t) &= P_0 e^{rt} \\ P_0 &= P(t) \, e^{-rt} \\ dI &= I(t) \, dt \\ dPV &= e^{-rt} \, dI = I(t) \, e^{-rt} \, dt \\ \end{aligned} \]

Average value of a function

\[ \begin{aligned} \overline{f} = \frac{\int_{x=a}^b f(x) \, dx}{\int_{x=a}^b dx} \end{aligned} \]

Note the average value over a region is the integral of the function over the region divided by the volume of that region.

Centroids and Centers Of Mass

\[ \begin{aligned} Area = \int \int_R dx\,dy\\ \end{aligned} \]

Centroids

\[ \begin{aligned} \overline{x} = \frac{\int \int_R x\,dx\,dy}{\int \int_R dx\,dy}\\ \overline{y} = \frac{\int \int_R y\,dx\,dy}{\int \int_R dx\,dy}\\ \end{aligned} \]

Centroids using point mass

Given a complex region which consists of the union of simpler regions, there is a method for finding the centroid:

1. Find the centroid of each simple region;
2. Replace each region with a point mass at its centroid, where the mass is the area of the region;
3. Find the centroid of these point masses by taking a weighted average of their x and y coordinates.

Center of mass

\[ \begin{aligned} \overline{x} = \frac{\int \int_R \rho(x,y)\, x\,dx\,dy}{\int \int_R \rho(x,y)\, dx\,dy}\\ \overline{y} = \frac{\int \int_R \rho(x,y)\, y\,dx\,dy}{\int \int_R \rho(x,y)\, dx\,dy}\\ \end{aligned} \]

Moments and Gyrations

\[ \begin{aligned} \text{Inertia} &= r^2 M \\ dI &= r^2 dM \\ \end{aligned} \]

Fair Probability

\[ \begin{aligned} P(D) &= \frac{\text{Volume of } D}{\text{Total volume of all outcomes}}\\ \end{aligned} \]

Probability density function (PDF)

Probability element:

\[ \begin{aligned} dP &= \rho(x) dx \\ \end{aligned} \]

2 properties of a PDF:

1. \(\displaystyle \rho(x) > 0 \, \forall x \in D\)
2. \(\displaystyle \int_D \rho(x) \, dx = 1\)

Expectation and Variance

Expectation - 1st moment:

\[ \begin{aligned} E &= \int_D x \, dP \\ &= \int_D x \, \rho(x) \, dx \\ \end{aligned} \]

Variance - 2nd moment:

\[ \begin{aligned} V &= \int_D (x - E)^2 \, dP \\ &= \int_D x^2 \, dP - E^2 \\ \end{aligned} \]

Interpreted as mass, expectation is the centroid, variance the moment of inertia about the centroid, and standard deviation the radius of gyration from the centroid !

\[ \begin{aligned} E &= \frac{\int_D x \, dP}{\int_D \, dP} \,\, \Leftrightarrow \, \, \overline x = \frac{\int x \, dM}{\int \, dM} \\ V &= \int_D (x - E)^2 \, dP \,\, \Leftrightarrow \, \, I = \int r^2 \, dM \\ \sigma &= \sqrt{\frac{\int_D (x - E)^2 \, dP}{\int_D \, dP}} = \sqrt V \,\, \Leftrightarrow \, \, R_g = \sqrt{\frac{I}{M}} = \sqrt{\frac{\int r^2 \, dM}{\int \, dM}} \\ \end{aligned} \]

Simplest Ordinary Differential Equation:

\[ \begin{aligned} \frac{dx}{dt} = ax \text{ , then } x &= Ce^{at} \end{aligned} \]

Linear 1st order differential equation:

\[ \begin{aligned} \frac{dx}{dt} = A(t)x + B(t) \end{aligned} \]

Standard form:

\[ \begin{aligned} \frac{dx}{dt} - Ax = B \end{aligned} \]

Integrating factor:

Used for solving linear 1st order differential equations via the product rule to factor the sum of two derivatives into the derivative of a product.

\[ \begin{aligned} I &= e^{\int{-A}} \\ \text{i.e. } I &= e^{\int{-A(t)dt}} \end{aligned} \]

such that:

\[ \begin{aligned} I\frac{dx}{dt} - IAx &= IB = \frac{d}{dt}Ix \\ \end{aligned} \]

Solution:

\[ \begin{aligned} x &= e^{\int{A}} \cdot \int{B e^{\int{-A}}} \\ \text{i.e. } x &= e^{\int{A(t)dt}} \cdot \int{B(t) e^{\int{-A(t)dt}}} \\ \end{aligned} \]

Equilibrium of \(\displaystyle \frac{dx}{dt} = f(x)\)

are simply the roots of \(f(x)\). Note if \(f'(x) < 0\), the equilibrium is stable. If \(f'(x) > 0\), the equilibrium is unstable.

Partial Fractions

\[ \begin{aligned} \frac{P(x)}{Q(x)} = \frac{A_1}{x-r_1} + \frac{A_2}{x-r_2} + \cdots + \frac{A_n}{x-r_n} \end{aligned} \]

Fundamental Theorem of Integral Calculus

\[ \begin{aligned} \frac{d}{dx} \int_{t=a}^x f(t)dt = f(x) \end{aligned} \]

p-integral

\( \displaystyle \int \frac{1}{x^p} dx = \int x^{-p} dx \)

Note

\(\displaystyle \int_1^\infty x^{-p}dx \)

diverges if \(p \le 1\) and converges otherwise; whereas

\(\displaystyle \int_0^1 x^{-p}dx \)

diverges if \(p \ge 1\) and converges otherwise.

For more info, check out the PennCalcWiki.