Geometry and Physical Applications
Cheat sheet.

The key rules, formulas and reminders from all 4 topics, gathered into reference cards.

Key formulas, conditions and traps · Read down each column.

Area Between Curves and Average Values

A=\int_a^b|f-g|dx,

f_{\rm avg}=\frac{\int_a^bf(x)dx}{b-a}.

Split where boundaries cross. Area has square units; an average has the same units as the original function.

Washers: $V=\pi\int(R^2-r^2)$ . Shells: $V=2\pi\int(\text{radius})(\text{height})$ .

Measure radii from the actual axis. Match the slice variable to the bounds and avoid overlapping shells.

ds=\sqrt{1+(y')^2}\,dx,\quad L=\int ds,\quad S=\int2\pi r\,ds.

Surface area is not volume and excludes end caps unless added separately. Radius is a nonnegative distance.

W=\int Fdx,\quad m=\int\lambda dx,\quad \bar x=\frac{\int x\lambda dx}{m}.

Use consistent distance units. Weight includes gravity; mass does not. A center of mass requires positive total mass.

2 reference blocks

A=\int_a^b|f-g|dx,

f_{\rm avg}=\frac{\int_a^bf(x)dx}{b-a}.

Split where boundaries cross. Area has square units; an average has the same units as the original function.

2 reference blocks

Washers: $V=\pi\int(R^2-r^2)$ . Shells: $V=2\pi\int(\text{radius})(\text{height})$ .

Measure radii from the actual axis. Match the slice variable to the bounds and avoid overlapping shells.

2 reference blocks

ds=\sqrt{1+(y')^2}\,dx,\quad L=\int ds,\quad S=\int2\pi r\,ds.

Surface area is not volume and excludes end caps unless added separately. Radius is a nonnegative distance.

2 reference blocks

W=\int Fdx,\quad m=\int\lambda dx,\quad \bar x=\frac{\int x\lambda dx}{m}.

Use consistent distance units. Weight includes gravity; mass does not. A center of mass requires positive total mass.