Let G G G f f f G G G G G G ( x k ∗ , y k ∗ , z k ∗ ) (x_k^*, y_k^*, z_k^*) ( x k ∗ , y k ∗ , z k ∗ ) Δ V k \Delta V_k Δ V k f f f G G G
∭ G f d V = lim n → ∞ ∑ k = 1 n f ( x k ∗ , y k ∗ , z k ∗ ) Δ V k \iiint_G f\,\text{d}V = \lim_{n \to \infty} \sum_{k=1}^{n} f(x_k^*, y_k^*, z_k^*)\,\Delta V_k ∭ G f d V = n → ∞ lim k = 1 ∑ n f ( x k ∗ , y k ∗ , z k ∗ ) Δ V k provided the limit exists.
Properties
Suppose f f f g g g G G G c ∈ R c \in \mathbb{R} c ∈ R
∭ G 1 d V \displaystyle \iiint_G 1\,\text{d}V ∭ G 1 d V G G G ∭ G f d V \displaystyle\iiint_G f\,\text{d}V ∭ G f d V G G G f f f ∭ G c f d V = c ∭ G f d V \displaystyle\iiint_G cf\,\text{d}V = c\iiint_G f\,\text{d}V ∭ G c f d V = c ∭ G f d V ∭ G ( f + g ) d V = ∭ G f d V + ∭ G g d V \displaystyle\iiint_G (f+g)\,\text{d}V = \iiint_G f\,\text{d}V + \iiint_G g\,\text{d}V ∭ G ( f + g ) d V = ∭ G f d V + ∭ G g d V If G = G 1 ∪ G 2 G = G_1 \cup G_2 G = G 1 ∪ G 2 G 1 G_1 G 1 G 2 G_2 G 2 ∭ G f d V = ∭ G 1 f d V + ∭ G 2 f d V \displaystyle\iiint_G f\,\text{d}V = \iiint_{G_1} f\,\text{d}V + \iiint_{G_2} f\,\text{d}V ∭ G f d V = ∭ G 1 f d V + ∭ G 2 f d V
Fubini’s Theorem
Let G G G a ≤ x ≤ b a \le x \le b a ≤ x ≤ b c ≤ y ≤ d c \le y \le d c ≤ y ≤ d k ≤ z ≤ l k \le z \le l k ≤ z ≤ l
If f f f G G G
∭ G f ( x , y , z ) d V = ∫ a b ∫ c d ∫ k l f ( x , y , z ) d z d y d x \iiint_G f(x,y,z)\,\text{d}V = \int_a^b \int_c^d \int_k^l f(x,y,z)\,\text{d}z\,\text{d}y\,\text{d}x ∭ G f ( x , y , z ) d V = ∫ a b ∫ c d ∫ k l f ( x , y , z ) d z d y d x The iterated integral can be written in any of the 5 other orders obtained by permuting the order of integrations (limits and d x \text{d}x d x d y \text{d}y d y d z \text{d}z d z
Regions of Integration
A solid region G G G x y xy x y iff :
Bounded above by z = v ( x , y ) z = v(x,y) z = v ( x , y ) z = u ( x , y ) z = u(x,y) z = u ( x , y )
u u u v v v R R R G G G x y xy x y u ( x , y ) ≤ v ( x , y ) u(x,y) \le v(x,y) u ( x , y ) ≤ v ( x , y ) ( x , y ) ∈ R (x,y) \in R ( x , y ) ∈ R
For f f f G G G
∭ G f ( x , y , z ) d V = ∬ R ( ∫ u ( x , y ) v ( x , y ) f ( x , y , z ) d z ) d A \iiint_G f(x,y,z)\,\text{d}V = \iint_R \left( \int_{u(x,y)}^{v(x,y)} f(x,y,z)\,\text{d}z \right) \text{d}A ∭ G f ( x , y , z ) d V = ∬ R ( ∫ u ( x , y ) v ( x , y ) f ( x , y , z ) d z ) d A The projection R R R
Type I, or vertically simple
R = { ( x , y ) : a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x ) } R = \{(x,y) : a \le x \le b,\ g_1(x) \le y \le g_2(x)\} R = {( x , y ) : a ≤ x ≤ b , g 1 ( x ) ≤ y ≤ g 2 ( x )}
Type II, or horizontally simple
R = { ( x , y ) : c ≤ y ≤ d , h 1 ( y ) ≤ x ≤ h 2 ( y ) } R = \{(x,y) : c \le y \le d,\ h_1(y) \le x \le h_2(y)\} R = {( x , y ) : c ≤ y ≤ d , h 1 ( y ) ≤ x ≤ h 2 ( y )}
If R R R
∭ G f ( x , y , z ) d V = ∫ a b ∫ g 1 ( x ) g 2 ( x ) ∫ u ( x , y ) v ( x , y ) f ( x , y , z ) d z d y d x \iiint_G f(x,y,z)\,\text{d}V = \int_a^b \int_{g_1(x)}^{g_2(x)} \int_{u(x,y)}^{v(x,y)} f(x,y,z)\,\text{d}z\,\text{d}y\,\text{d}x ∭ G f ( x , y , z ) d V = ∫ a b ∫ g 1 ( x ) g 2 ( x ) ∫ u ( x , y ) v ( x , y ) f ( x , y , z ) d z d y d x If R R R
∭ G f ( x , y , z ) d V = ∫ c d ∫ h 1 ( y ) h 2 ( y ) ∫ u ( x , y ) v ( x , y ) f ( x , y , z ) d z d x d y \iiint_G f(x,y,z)\,\text{d}V = \int_c^d \int_{h_1(y)}^{h_2(y)} \int_{u(x,y)}^{v(x,y)} f(x,y,z)\,\text{d}z\,\text{d}x\,\text{d}y ∭ G f ( x , y , z ) d V = ∫ c d ∫ h 1 ( y ) h 2 ( y ) ∫ u ( x , y ) v ( x , y ) f ( x , y , z ) d z d x d y Analogous iterated integrals hold if G G G y z yz y z z x zx z x
Cylindrical Coordinates
Cylindrical coordinates are defined by
x = r cos θ , y = r sin θ , z = z , r ≥ 0 x = r\cos\theta, \quad y = r\sin\theta, \quad z = z, \quad r \ge 0 x = r cos θ , y = r sin θ , z = z , r ≥ 0 The volume element is
d V = r d z d r d θ \text{d}V = r\,\text{d}z\,\text{d}r\,\text{d}\theta d V = r d z d r d θ Let G G G z = v ( r , θ ) z = v(r,\theta) z = v ( r , θ ) z = u ( r , θ ) z = u(r,\theta) z = u ( r , θ ) x y xy x y R ′ R' R ′ r θ r\theta r θ f f f G G G
∭ G f ( x , y , z ) d V = ∬ R ′ ( ∫ u ( r , θ ) v ( r , θ ) f ( r cos θ , r sin θ , z ) d z ) r d A \iiint_G f(x,y,z)\,\text{d}V = \iint_{R'} \left( \int_{u(r,\theta)}^{v(r,\theta)} f(r\cos\theta, r\sin\theta, z)\,\text{d}z \right) r\,\text{d}A ∭ G f ( x , y , z ) d V = ∬ R ′ ( ∫ u ( r , θ ) v ( r , θ ) f ( r cos θ , r sin θ , z ) d z ) r d A If for each fixed θ ∈ [ α , β ] \theta \in [\alpha, \beta] θ ∈ [ α , β ] r r r r 1 ( θ ) r_1(\theta) r 1 ( θ ) r 2 ( θ ) r_2(\theta) r 2 ( θ )
∭ G f ( x , y , z ) d V = ∫ α β ∫ r 1 ( θ ) r 2 ( θ ) ∫ u ( r , θ ) v ( r , θ ) f ( r cos θ , r sin θ , z ) r d z d r d θ \iiint_G f(x,y,z)\,\text{d}V = \int_\alpha^\beta \int_{r_1(\theta)}^{r_2(\theta)} \int_{u(r,\theta)}^{v(r,\theta)} f(r\cos\theta, r\sin\theta, z)\,r\,\text{d}z\,\text{d}r\,\text{d}\theta ∭ G f ( x , y , z ) d V = ∫ α β ∫ r 1 ( θ ) r 2 ( θ ) ∫ u ( r , θ ) v ( r , θ ) f ( r cos θ , r sin θ , z ) r d z d r d θ Spherical Coordinates
Spherical coordinates are defined by
x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ x = \rho\sin\phi\cos\theta, \quad y = \rho\sin\phi\sin\theta, \quad z = \rho\cos\phi x = ρ sin ϕ cos θ , y = ρ sin ϕ sin θ , z = ρ cos ϕ
ρ \rho ρ θ \theta θ x y xy x y x x x ϕ \phi ϕ z z z
Standard ranges are ρ ≥ 0 \rho \ge 0 ρ ≥ 0 0 ≤ θ ≤ 2 π 0 \le \theta \le 2\pi 0 ≤ θ ≤ 2 π 0 ≤ ϕ ≤ π 0 \le \phi \le \pi 0 ≤ ϕ ≤ π z z z ρ sin ϕ \rho\sin\phi ρ sin ϕ
The volume element is
d V = ρ 2 sin ϕ d ρ d ϕ d θ \text{d}V = \rho^2 \sin\phi\,\text{d}\rho\,\text{d}\phi\,\text{d}\theta d V = ρ 2 sin ϕ d ρ d ϕ d θ For f f f G G G
∭ G f ( x , y , z ) d V = ∭ f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) ρ 2 sin ϕ d ρ d ϕ d θ \iiint_G f(x,y,z)\,\text{d}V = \iiint f(\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)\,\rho^2\sin\phi\,\text{d}\rho\,\text{d}\phi\,\text{d}\theta ∭ G f ( x , y , z ) d V = ∭ f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) ρ 2 sin ϕ d ρ d ϕ d θ Common Spherical Regions
Spherical shell between ρ = ρ 1 \rho = \rho_1 ρ = ρ 1 ρ = ρ 2 \rho = \rho_2 ρ = ρ 2 ρ 1 < ρ 2 \rho_1 < \rho_2 ρ 1 < ρ 2 ρ 1 ≤ ρ ≤ ρ 2 \rho_1 \le \rho \le \rho_2 ρ 1 ≤ ρ ≤ ρ 2 0 ≤ ϕ ≤ π 0 \le \phi \le \pi 0 ≤ ϕ ≤ π 0 ≤ θ ≤ 2 π 0 \le \theta \le 2\pi 0 ≤ θ ≤ 2 π
Sphere ρ = ρ 0 \rho = \rho_0 ρ = ρ 0 0 ≤ ρ ≤ ρ 0 0 \le \rho \le \rho_0 0 ≤ ρ ≤ ρ 0 0 ≤ ϕ ≤ π 2 0 \le \phi \le \frac{\pi}{2} 0 ≤ ϕ ≤ 2 π 0 ≤ θ ≤ π 2 0 \le \theta \le \frac{\pi}{2} 0 ≤ θ ≤ 2 π
Sphere ρ = ρ 0 \rho = \rho_0 ρ = ρ 0 ϕ = ϕ 0 \phi = \phi_0 ϕ = ϕ 0 0 ≤ ρ ≤ ρ 0 0 \le \rho \le \rho_0 0 ≤ ρ ≤ ρ 0 0 ≤ ϕ ≤ ϕ 0 0 \le \phi \le \phi_0 0 ≤ ϕ ≤ ϕ 0 0 ≤ θ ≤ 2 π 0 \le \theta \le 2\pi 0 ≤ θ ≤ 2 π
Sphere ρ = ρ 0 \rho = \rho_0 ρ = ρ 0 ϕ = ϕ 1 \phi = \phi_1 ϕ = ϕ 1 ϕ = ϕ 2 \phi = \phi_2 ϕ = ϕ 2 ϕ 1 < ϕ 2 \phi_1 < \phi_2 ϕ 1 < ϕ 2 0 ≤ ρ ≤ ρ 0 0 \le \rho \le \rho_0 0 ≤ ρ ≤ ρ 0 ϕ 1 ≤ ϕ ≤ ϕ 2 \phi_1 \le \phi \le \phi_2 ϕ 1 ≤ ϕ ≤ ϕ 2 0 ≤ θ ≤ 2 π 0 \le \theta \le 2\pi 0 ≤ θ ≤ 2 π
Solid enclosed laterally by the cone ϕ = ϕ 0 \phi = \phi_0 ϕ = ϕ 0 0 < ϕ 0 < π 2 0 < \phi_0 < \frac{\pi}{2} 0 < ϕ 0 < 2 π z = a z = a z = a a > 0 a > 0 a > 0 0 ≤ ρ ≤ a sec ϕ 0 \le \rho \le a\sec\phi 0 ≤ ρ ≤ a sec ϕ 0 ≤ ϕ ≤ ϕ 0 0 \le \phi \le \phi_0 0 ≤ ϕ ≤ ϕ 0 0 ≤ θ ≤ 2 π 0 \le \theta \le 2\pi 0 ≤ θ ≤ 2 π