Key Equations

Double integral	$\iint_{R} f (x, y) d A = lim_{m, n \to \infty} \sum_{i = 1}^{m} \sum_{j = 1}^{n} f (x_{i j}^{}, y_{i j}^{}) Δ A$
Iterated integral	$\int_{a}^{b} \int_{c}^{d} f (x, y) d x d y = \int_{a}^{b} [\int_{c}^{d} f (x, y) d y] d x$ or $\int_{c}^{d} \int_{b}^{a} f (x, y) d x d y = \int_{c}^{d} [\int_{a}^{b} f (x, y) d x] d y$
Average value of a function of two variables	$f_{ave} = \frac{1}{Area R} \iint_{R} f (x, y) d x d y$

Iterated integral over a Type I region	$\iint_{D} f (x, y) d A = \iint_{D} f (x, y) d y d x = \int_{a}^{b} [\int_{g_{1} (x)}^{g_{2} (x)} f (x, y) d y] d x$
Iterated integral over a Type II region	$\iint_{D} f (x, y) d A = \iint_{D} f (x, y) d x d y = \int_{c}^{d} [\int_{h_{1} (y)}^{h_{2} (y)} f (x, y) d x] d y$

Double integral over a polar rectangular region $R$	$\iint_{R} f (r, θ) d A = lim_{m, n \to \infty} \sum_{i = 1}^{m} \sum_{j = 1}^{n} f (r_{i j} , θ_{i j} ) Δ A = lim_{m, n \to \infty} \sum_{i = 1}^{m} \sum_{j = 1}^{n} f (r_{i j} , θ_{i j} ) r_{i j} * Δ r Δ θ$
Double integral over a general polar region	$\iint_{D} f (r, θ) r d r d θ = \int_{θ = α}^{θ = β} \int_{r = h_{1} (θ)}^{r = h_{2} (θ)} f (r, θ) r d r d θ$

Triple integral

lim_{l, m, n \to \infty} \sum_{i = 1}^{l} \sum_{j = 1}^{m} \sum_{k = 1}^{n} f (x_{i j k}^{*}, y_{i j k}^{*}, z_{i j k}^{*}) Δ x Δ y Δ z = \underset{B}{∭} f (x, y, z) d V

Triple integral in cylindrical coordinates	$\underset{B}{∭} g (x, y, z) d V = \underset{B}{∭} g (r cos θ, r sin θ, z) r d r d θ d z = \underset{B}{∭} f (r, θ, z) r d r d θ d z$
Triple integral in spherical coordinates	$\underset{B}{∭} f (ρ, θ, φ) ρ^{2} sin φ d ρ d φ d θ = \int_{φ = γ}^{φ = ψ} \int_{θ = α}^{θ = β} \int_{ρ = a}^{ρ = b} f (ρ, θ, φ) ρ^{2} sin φ d ρ d φ d θ$

Mass of a lamina	$m = lim_{k, l \to \infty} \sum_{i = 1}^{k} \sum_{j = 1}^{l} m_{i j} = lim_{k, l \to \infty} \sum_{i = 1}^{k} \sum_{j = 1}^{l} ρ (x_{i j}^{}, y_{i j}^{}) Δ A = \iint_{R} ρ (x, y) d A$
Moment about the x-axis	$M_{x} = lim_{k, l \to \infty} \sum_{i = 1}^{k} \sum_{j = 1}^{l} (y_{i j}^{}) m_{i j} = lim_{k, l \to \infty} \sum_{i = 1}^{k} \sum_{j = 1}^{l} (y_{i j}^{}) ρ (x_{i j}^{}, y_{i j}^{}) Δ A = \iint_{R} y ρ (x, y) d A$
Moment about the y-axis	$M_{y} = lim_{k, l \to \infty} \sum_{i = 1}^{k} \sum_{j = 1}^{l} (x_{i j}^{}) m_{i j} = lim_{k, l \to \infty} \sum_{i = 1}^{k} \sum_{j = 1}^{l} (x_{i j}^{}) ρ (x_{i j}^{}, y_{i j}^{}) Δ A = \iint_{R} x ρ (x, y) d A$
Center of mass of a lamina	$\overset{−}{x} = \frac{M_{y}}{m} = \frac{\iint_{R} x ρ (x, y) d A}{\iint_{R} ρ (x, y) d A}$ and $\overset{−}{y} = \frac{M_{x}}{m} = \frac{\iint_{R} y ρ (x, y) d A}{\iint_{R} ρ (x, y) d A}$