APEX Additional Formulas

Section B.6 Additional Formulas

\begin{aligned} \sum_{i = 1}^{n} c & = c n & \sum_{i = 1}^{n} i & = \frac{n (n + 1)}{2} \\ \sum_{i = 1}^{n} i^{2} & = \frac{n (n + 1) (2 n + 1)}{6} & \sum_{i = 1}^{n} i^{3} & = {(\frac{n (n + 1)}{2})}^{2} \end{aligned}

\int_{a}^{b} f (x) d x \approx \frac{Δ x}{2} [f (x_{0}) + 2 f (x_{1}) + 2 f (x_{2}) + \dots + 2 f (x_{n - 1}) + f (x_{n})]

with Error

\leq \frac{(b - a)^{3}}{12 n^{2}} [max | f^{″} (x) |]

\int_{a}^{b} f (x) d x \approx \frac{Δ x}{3} [f (x_{0}) + 4 f (x_{1}) + 2 f (x_{2}) + 4 f (x_{3}) + \dots + 2 f (x_{n - 2}) + 4 f (x_{n - 1}) + f (x_{n})]

with Error

\leq \frac{(b - a)^{5}}{180 n^{4}} [max | f^{(4)} (x) |]

L = \int_{a}^{b} \sqrt{1 + f^{'} (x)^{2}} d x

2 π \int_{a}^{b} f (x) \sqrt{1 + f^{'} (x)^{2}} d x

(where

f (x) \geq 0

)

S = 2 π \int_{a}^{b} x \sqrt{1 + f^{'} (x)^{2}} d x

(where

a, b \geq 0

)

W = \int_{a}^{b} F (x) d x

F = \int_{a}^{b} w d (y) ℓ (y) d y

p_{n} (x) = f (c) + f^{'} (c) (x - c) + \frac{f^{″} (c)}{2!} (x - c)^{2} + \dots + \frac{f^{(n)} (c)}{n!} (x - c)^{n} + \dots

p_{n} (x) = f (0) + f^{'} (0) x + \frac{f^{″} (0)}{2!} x^{2} + \frac{f^{‴} (0)}{3!} x^{3} + \dots + \frac{f^{(n)} (0)}{n!} x^{n} + \dots

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