← Advanced Calculus
Mathematics Β· Topic Cheatsheet

Advanced Calculus

17 key results accumulated across 2 chapters.

Area under a curve
Ch 1
A=∫abf(x) dxA = \int_a^b f(x)\,dx
Volume of revolution (x-axis)
Ch 1
V=Ο€βˆ«ab[f(x)]2 dxV = \pi \int_a^b [f(x)]^2\,dx
Kinematics links
Ch 1
v=dsdt,a=dvdt,s=∫v dtv = \frac{ds}{dt},\quad a = \frac{dv}{dt},\quad s = \int v\,dt
Signed area
Ch 1
Area below the x-axis counts negative; for geometric area integrate regions separately.
Area between curves
Ch 1
∫ab(topβˆ’bottom) dx\int_a^b (\text{top}-\text{bottom})\,dx
Volume of revolution
Ch 1
V=Ο€βˆ«aby2 dxV = \pi\int_a^b y^2\,dx
Kinematics (calculus)
Ch 1
v=dsdt,β€…β€Ša=dvdt,β€…β€Šs=∫v dtv=\tfrac{ds}{dt},\; a=\tfrac{dv}{dt},\; s=\int v\,dt
Common trap
Ch 1
Displacement = ∫v dt (signed); distance = ∫|v| dt (split where v changes sign).
Separation of variables
Ch 2
dydx=f(x)g(y)β‡’βˆ«dyg(y)=∫f(x) dx\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)\,dx
Exponential growth/decay
Ch 2
dydx=ky⇒y=Aekx\frac{dy}{dx}=ky \Rightarrow y = Ae^{kx}
Constant AA fixed by the initial condition.
Maclaurin series
Ch 2
f(x)=βˆ‘n=0∞f(n)(0)n!xnf(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n
Standard series
Ch 2
ex=βˆ‘xnn!,β€…β€Šsin⁑x=xβˆ’x33!+⋯ ,β€…β€Šcos⁑x=1βˆ’x22!+β‹―e^x=\sum\frac{x^n}{n!},\;\sin x=x-\tfrac{x^3}{3!}+\cdots,\;\cos x=1-\tfrac{x^2}{2!}+\cdots
ln(1+x)
Ch 2
ln⁑(1+x)=xβˆ’x22+x33βˆ’β‹―\ln(1+x)=x-\tfrac{x^2}{2}+\tfrac{x^3}{3}-\cdots
Valid for βˆ’1<x≀1-1<x\le 1.
Maclaurin
Ch 2
f(x)=βˆ‘f(n)(0)n!xnf(x)=\sum \tfrac{f^{(n)}(0)}{n!}x^n
Key expansions
Ch 2
ex=βˆ‘xnn!,β€…β€Šsin⁑x=xβˆ’x33!+β‹―e^x=\sum\tfrac{x^n}{n!},\;\sin x = x-\tfrac{x^3}{3!}+\cdots
Separable ODE
Ch 2
∫dyg(y)=∫f(x) dx\int\tfrac{dy}{g(y)} = \int f(x)\,dx
Separate variables, integrate both sides, +C from initial condition.
Common trap
Ch 2
Apply the initial condition to find C; Euler’s method is an approximation (step size matters).