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Advanced Calculus

Chapter 1 · Areas, Volumes, Motion

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Area under a curve
A=∫abf(x) dxA = \int_a^b f(x)\,dx
Volume of revolution (x-axis)
V=π∫ab[f(x)]2 dxV = \pi \int_a^b [f(x)]^2\,dx
Kinematics links
v=dsdt,a=dvdt,s=∫v dtv = \frac{ds}{dt},\quad a = \frac{dv}{dt},\quad s = \int v\,dt
Signed area
Area below the x-axis counts negative; for geometric area integrate regions separately.
Area between curves
∫ab(top−bottom) dx\int_a^b (\text{top}-\text{bottom})\,dx
Volume of revolution
V=π∫aby2 dxV = \pi\int_a^b y^2\,dx
Kinematics (calculus)
v=dsdt,  a=dvdt,  s=∫v dtv=\tfrac{ds}{dt},\; a=\tfrac{dv}{dt},\; s=\int v\,dt
Common trap
Displacement = ∫v dt (signed); distance = ∫|v| dt (split where v changes sign).