Double Integrals
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.

Double Integrals and Accumulation

\iint_Rf\,dA=\lim\sum f(x_i,y_i)\Delta A_i,

\bar f=\frac{\iint_Rf\,dA}{\operatorname{area}(R)}.

An inner integration treats the other variable as fixed. Signed accumulation, absolute volume and average value are different quantities.

Describe the region as inequalities, sketch its slices, then solve for bounds in the desired order.

Changing integration order changes bounds as well as differentials. Split the region where its active boundary changes.

x=r\cos\theta,\quad y=r\sin\theta,\quad dA=r\,dr\,d\theta.

Transform the region, integrand and area element together. Avoid tracing a region more than once.

dA=\left|\det\frac{\partial(x,y)}{\partial(u,v)}\right|du dv.

Use the absolute forward Jacobian for area. Check one-to-one coverage and distinguish forward from inverse determinants.

2 reference blocks

\iint_Rf\,dA=\lim\sum f(x_i,y_i)\Delta A_i,

\bar f=\frac{\iint_Rf\,dA}{\operatorname{area}(R)}.

An inner integration treats the other variable as fixed. Signed accumulation, absolute volume and average value are different quantities.

2 reference blocks

Describe the region as inequalities, sketch its slices, then solve for bounds in the desired order.

Changing integration order changes bounds as well as differentials. Split the region where its active boundary changes.

2 reference blocks

x=r\cos\theta,\quad y=r\sin\theta,\quad dA=r\,dr\,d\theta.

Transform the region, integrand and area element together. Avoid tracing a region more than once.

2 reference blocks

dA=\left|\det\frac{\partial(x,y)}{\partial(u,v)}\right|du dv.

Use the absolute forward Jacobian for area. Check one-to-one coverage and distinguish forward from inverse determinants.