# APEX Calculus

### Summation Formulas:.

\begin{align*} \sum^n_{i=1}{c} \amp = cn \amp \sum^n_{i=1}{i} \amp= \frac{n(n+1)}{2}\\ \sum^n_{i=1}{i^2} \amp = \frac{n(n+1)(2n+1)}{6} \amp \sum^n_{i=1}{i^3} \amp = \left(\frac{n(n+1)}{2}\right)^2 \end{align*}

### Trapezoidal Rule:.

\begin{equation*} \int_a^b{f(x)}\, dx \approx \frac{\Delta x}{2}\big[f(x_0)+2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_{n})\big] \end{equation*}
with Error $$\ds \leq \frac{(b-a)^3}{12n^2}\big[\max \abs{f\,''(x)}\big]$$

### Simpson’s Rule:.

\begin{equation*} \int_a^b{f(x)}\, dx \approx \frac{\Delta x}{3}\big[f(x_0)+4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_{n})\big] \end{equation*}
with Error $$\ds \leq \frac{(b-a)^5}{180n^4}\big[ \max\abs{f\,^{(4)}(x)}\big]$$

### Arc Length:.

\begin{equation*} L = \int_a^b\sqrt{1+ f\,'(x)^2}\,dx \end{equation*}

### Surface of Revolution:.

\begin{equation*} 2\pi \int_a^b{f(x) \sqrt{1+ f\,'(x)^2}}dx \end{equation*}
(where $$f(x)\geq 0$$)
\begin{equation*} S = 2\pi \int_a^b{x \sqrt{1+ f\,'(x)^2}}dx \end{equation*}
(where $$a,b \geq 0$$)

### Work Done by a Variable Force:.

\begin{equation*} W = \int_a^b{F(x)}dx \end{equation*}

### Force Exerted by a Fluid:.

\begin{equation*} F = \int_a^b{w\,d(y)\,\ell(y)}dy \end{equation*}

### Taylor Series Expansion for $$f(x)\text{:}$$.

\begin{equation*} p_n(x) = f(c) + f\,'(c)(x-c) + \frac{f\,''(c)}{2!}(x-c)^2 + \cdots + \frac{f\,^{(n)}(c)}{n!}(x-c)^n + \cdots \end{equation*}

### Maclaurin Series Expansion for $$f(x)\text{,}$$ where $$c=0\text{:}$$.

\begin{equation*} p_n(x) = f(0) + f\,'(0)x + \frac{f\,''(0)}{2!}x^2 + \frac{f\,'''(0)}{3!}x^3 + \cdots + \frac{f\,^{(n)}(0)}{n!}x^n+\cdots \end{equation*}