微積分上課筆記1

微積分上課筆記

• Euler method 開方

Volumes by Cylinderical Shells

[time=2022,11,14]

Disk method

$\int _a ^b \pi f^2(x)dx$

Shell method

$\int^{d}_{c}2\pi y f(y)dy$

check

算球體體積

Calc the volume of a ball

這是半球體積

$r = x$
$h=\sqrt{1^2 - x^2}$
$Volume = \int_0^1 2\pi x\sqrt{1^2 - x^2}dx$
$let$ $u = 1 - x^2$
$V = \pi\int_0^1\sqrt u$ $du$
$= \pi \frac{2}{3}(u) ^{\frac{3}{2}}|_{u = 0}^{1}$
$= \frac{\frac{4}{3}\pi 1^3}{2} = \frac{2}{3}$

Mean Value Theorem for Integrals

均值定理

If $f(x)$ is continuous on $[a,b]$, then there exists a number $c$ in $[a,b]$ such that
$f(c)=f_{ave}=\frac{1}{b - a}\int_a^bf(x)dx$ that is, $\int^b_af(x)dx=f(c)(b - a)$

例題

Proof that there is at least one root in interval $[0,\pi]$ in $f(x)= cosx + 2cos(2x) + … + ncos(nx)$

solution

$\int_0^\pi f(x)dx$

$=\int_0^\pi \sum^{n}_{k=1} kcos(kx) dx$

$=\sum^{n}_{k=1}sin(kx)|_0^\pi$

$=0=f(c)(\pi - 0)$

Functions and Inverse Functions

只有一對一函數有
$f(x) = \sqrt{x} \rightarrow f^{-1}(x) = y^2$