Chapter 8 Formulas

Chapter 8
Integral Calculus and Its Uses

Chapter Summary

Formulas

Integration techniques based on derivative formulas

Power Rule

\int u^{n} d u = \frac{1}{n + 1} u^{n + 1} + C, (n \neq - 1)

Exception to the Power Rule

\int u^{- 1} d u = \ln | u | + C

Exponential Rules

\int e^{u} d u = e^{u} + C

\int a^{u} d u = \frac{1}{\ln a} a^{u} + C

Trigonometric Rules

\int \sin u d u = - \cos u + C

\int \cos u d u = \sin u + C

\int \sec^{2} u d u = \tan u + C

Substitution in an indefinite integral (Chain Rule)

\int f (u (x)) u^{'} (x) d x = \int f (u) d u

Substitution in a definite integral (Chain Rule)

\int_{a}^{b} f (u (x)) u^{'} (x) d x = \int_{u (a)}^{u (b)} f (u) d u

Integration by parts (Product Rule)

\int u d v = u v - \int v d u

Centers of mass

For a uniform tapered rod lying between $a$ and $b$ on the $x$-axis with cross-sectional radius $r(x)$:

\bar{x} = \frac{\int_{a}^{b} x [{r (x)}^{2}] d x}{\int_{a}^{b} [{r (x)}^{2}] d x}

For a uniform thin plate described by $a \leq x \leq b$ and $g(x) \leq y \leq f(x)$:

\bar{x} = \frac{\int_{a}^{b} x [f (x) - g (x)] d x}{\int_{a}^{b} [f (x) - g (x)] d x}

\bar{y} = \frac{1}{2} \frac{\int_{a}^{b} [{f (x)}^{2} - {g (x)}^{2}] d x}{\int_{a}^{b} [f (x) - g (x)] d x} .

Numerical approximations to the definite integral $\int_a^{\,b}f(x)\,dx$

In each of the following formulas the function $f$ is continuous on the interval $[a,b]$, $n$ is a positive integer, $\Delta t = \frac{b-a}{n}$, and $t_k = a + k \Delta t$ for $k = 0,\,1,\,...,\,n$.

Left-Hand Sum

LHS = \sum_{k = 1}^{n} f (t_{k - 1}) Δ t

Right-Hand Sum

RHS = \sum_{k = 1}^{n} f (t_{k}) Δ t

Trapezoidal Rule

TR = \sum_{k = 1}^{n} \frac{f (t_{k - 1}) + f (t_{k})}{2} Δ t

Midpoint Rule

MR = \sum_{k = 1}^{n} f (\frac{t_{k - 1} + t_{k}}{2}) Δ t

Simpson's Rule

SR = \frac{2}{3} MR + \frac{1}{3} TR

Fourier coefficients

$f(t) = b_0 + a_1 \sin\,t + b_1 \cos\,t + a_2 \sin\,2t + b_2 \cos\,2t + \cdots + a_n \sin\,nt + b_n \cos\,nt$

is a trigonometric polynomial, then the coefficients can be found by the following formulas:

$b_{0} = \frac{1}{2 π} \int_{- π}^{π} f (t) d t$

$b_{k} = \frac{1}{π} \int_{- π}^{π} f (t) \cos k t d t$ for $k = 1, 2, ..., n$

$a_{k} = \frac{1}{π} \int_{- π}^{π} f (t) \sin k t d t$ for $k = 1, 2, ..., n$

Contents for Chapter 8

Chapter 8 Integral Calculus and Its Uses

Chapter Summary

Chapter 8
Integral Calculus and Its Uses