 | Term | Definition |
| |
Volume of a solid region |
V = ∫∫f(x,y)dA |
| |
Fubini's Theorem |
∫∫f(x,y)dA = ∫∫f(x,y)dydx, for a≤x≤b and g₁(x)≤y≤g₂(x) or ∫∫f(x,y)dA = ∫∫f(x,y)dxdy, for c≤y≤d and h₁(x)≤x≤h₂(x) |
| |
double integral in polar coordinates |
∫∫f(r,θ)dA = ∫∫f(r,θ)rdrdθ |
| |
Polar form of Fubini's Theorem |
∫∫f(r,θ)dA = ∫∫f(r,θ)rdrdθ, for θ₁≤θ≤θ₂ and g₁(θ)≤y≤g₂(θ) or ∫∫f(r,θ)dA = ∫∫f(r,θ)rdθdr, for r₁≤r≤r₂ and h₁(θ)≤θ≤h₂(θ) |
| |
Change of Variables to Polar Form |
∫∫f(x,y)dA = ∫∫f(rcosθ, rsinθ)rdrdθ |
| |
Mass of a Planar Lamina of Variable Density |
m = ∫∫ρ(x,y)dA |
| |
Moments of mass of a variable density planar lamina |
M(x) = ∫∫yρ(x,y)dA and M(y) = ∫∫xρ(x,y)dA |
| |
center of mass of variable density planar lamina |
(x,y) = (M(x)/m, M(y)/m) |
| |
Moments of intertia of a variable density planar lamina |
M(x) = ∫∫y²ρ(x,y)dA and M(y) = ∫∫x²ρ(x,y)dA |
| |
Surface Area |
S = ∫∫ds = ∫∫√(1+[∂ƒ/∂x]² + [∂ƒ/∂y]²)dA where z =ƒ(x,y) |
| |
Evaluation of Iterated Integrals |
∫∫∫ƒ(x,y,z)dV = ∫∫∫ƒ(x,y,z)dzdy,dx where a≤x≤b, h₁(x)≤yh₂(x), g₁(x,y)≤z≤g₂(x,y) |
| |
Triple Integral In Cyclindrical Coordinates |
∫∫∫ƒ(r,θ,z)rdrdθdz |
| |
Triple Integral In Spherical Coordinates |
∫∫∫ƒ(ρ,θ,φ)ρ²sinφdrdθdφ |
| |
Jacobian |
∂(x,y)/∂(u,v) = |[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]| = (∂x/∂u)(∂y/∂v) - (∂y/∂u)(∂x/∂v) |
| |
Change of Variables for Double Integrals |
∫∫ƒ(x,y)dA = ∫∫ƒ(g(u,v),h(u,v)) |∂(x,y)/∂(u,v)| dudv |
Share on Facebook
Share on MySpace