Evaluating double integrals over rectangular intervals,
$\int_a^b\int_c^d f(x,y)\,dx\,dy$,
in Cartesian coordinates
using partial integration. In either order (Fubini's theorem).
Evaluating integrals like $$\int_a^b\int_{y=g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx.$$
Interpreting this integral as the integral over the area $a\lt x \lt b$ and between the curves $g_1(x)\lt y \lt g_2(x)$.
Order matters!
Being able to sketch the area of integration,
Adjusting the limits of integration to change the order of integration.
Double integrals in polar coordinates, like $\iint f(r,\theta)\,dA$.
Conversions between $x,y,z \leftrightarrow r,\theta,z$ coordinate systems.
Being able to set the limits on $r$ and $\theta$ from a sketch of the area of integration. Or, being able to interpret the limits on $r$ and $\theta$ integrals as an area of integration.
Using the double integral differential of area: $dA\equiv rdr\,d\theta$ when integrating over an area in 2-d.
Using an integral like $\iint dA=A$ to find the area of... the area of integration.
Triple integrals
Evaluating triple integrals over rectangular intervals,
$\int_a^b\int_c^d \int_g^h f(x,y,z)\,dx\,dy\,dz$,
in Cartesian coordinates
using partial integration. In either order (Fubini's theorem).
Evaluating integrals like $$\int_a^b\int_{y=g_1(x)}^{g_2(x)}\int_{z=h_1(x,y)}^{h_2(x,y)} f(x,y,z)\,dz\,dy\,dx.$$
Interpreting this integral in terms of a volume of integration running from $a\lt x \lt b$, between the curves $g_1(x)\lt y \lt g_2(x)$, and between the surfaces
$h_1(x,y)\lt z \lt h_2(x,y)$.
Order matters!
Being able to sketch or recognize the volume of integration,
Adjusting the limits of integration to change the order of integration.
Triple integrals in cylindrical coordinates, like $\iiint f(r,\theta,z)\,dv$.
Conversions between $x,y,z \leftrightarrow r,\theta,z$ coordinate systems.
Being able to set the limits on $r$, $\theta$, and $z$ from a sketch of the volume of integration. Or, being able to interpret the limits on $r$, $\theta$, and $z$ integrals as an volume of integration.
Using the triple integral differential of area: $dV\equiv rdr\,d\theta\,dz$ when integrating over a volume in 3-d.
Triple integrals in spherical-polar coordinates, like $\iiint f(\rho,\theta,\phi)\,dv$.
Conversions between $x,y,z \leftrightarrow r,\theta,\phi$ coordinate systems.
Being able to set the limits on $\rho$, $\theta$, and $\phi$ from a sketch of the volume of integration. Or, being able to interpret the limits on $\rho$, $\theta$, and $\phi$ integrals as an volume of integration.
Using the triple integral differential of area: $dV\equiv \rho^2dr\,d\theta\,\sin\phi d\phi$ when integrating over a volume in 3-d.
Using an integral like $\iiint dV=V$ to find the Volume of... the volume of integration.