Additional Formulas

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Subsection A.4 Additional Formulas

Summation Formulas.

\(\begin{array}{ll} \displaystyle \sum^n_{i=1}{c} = cn & \displaystyle \sum^n_{i=1}{i} = \frac{n(n+1)}{2} \\ \\ \displaystyle \sum^n_{i=1}{i\hspace{1.0pt}^2} = \frac{n(n+1)(2n+1)}{6} \quad& \displaystyle \sum^n_{i=1}{i\hspace{1.0pt}^3}= \left(\frac{n(n+1)}{2}\right)^2 \\ \end{array}\)

Trapezoidal Rule:.

\(\displaystyle \int _a^b{f(x)}\ dx \approx \frac{\Delta x}{2}\big [f(x_1)+2f(x_2) + 2f(x_3) + ... + 2f(x_{n}) + f(x_{n+1})\big ]\)

with \(\displaystyle \text{Error} \le \frac{(b-a)^3}{12n^2}\big [ \max \big | f^{\prime \prime }(x) \big |\big ]\)

Simpson's Rule.

\(\displaystyle \int _a^b{f(x)}\ dx \approx \frac{\Delta x}{3}\big [f(x_1)+4f(x_2) + 2f(x_3) + 4f(x_4) + ... + 2f(x_{n-1}) + 4f(x_{n}) + f(x_{n+1})\big ] \)

with \(\displaystyle \text{Error} \le \frac{(b-a)^5}{180n^4}\big [ \max \big | f^{(4)}(x) \big |\big ] \)

Arc Length.

\(\displaystyle L = \int _a^b{\sqrt{1+ f\,^{\prime }(x)^2}}\phantom{a}dx \)

Surface of Revolution.

\(\displaystyle S = 2\pi \int _a^b{f(x) \sqrt{1+ f\,^{\prime }(x)^2}}\phantom{a}dx \)

(where \(f(x)\ge 0\))

\(\displaystyle S = 2\pi \int _a^b{x \sqrt{1+ f\,^{\prime }(x)^2}}\phantom{a}dx \)

(where \(a,b \ge 0\))

Work Done by a Variable Force.

\(\displaystyle W = \int _a^b{F(x)}\phantom{a}dx \)

Force Exerted by a Fluid:.

\(\displaystyle F = \int _a^b{w\,d(y)\,\ell (y)}\phantom{a}dy \)

Taylor Series Expansion for \(f(x)\).

\(\displaystyle p_n(x) = f(c) + f\,^{\prime }(c)(x-c) + \frac{f\,^{\prime \prime }(c)}{2!}(x-c)^2 + \frac{f\,^{\prime \prime \prime }(c)}{3!}(x-c)^3 + ... + \frac{f\,^{(n)}(c)}{n!}(x-c)^n \)

Maclaurin Series Expansion for \(f(x)\).

\(\displaystyle p_n(x) = f(0) + f\,^{\prime }(0)x + \frac{f\,^{\prime \prime }(0)}{2!}x^2 + \frac{f\,^{\prime \prime \prime }(0)}{3!}x^3 + ... + \frac{f\,^{(n)}(0)}{n!}x^n \)