MATH 230 FORMULA SHEET (Ch. 14.1–14.7, 15.1–15.5)
DOUBLE INTEGRALS
■R f(x,y)dA = ∫a^b ∫c^d f(x,y) dy dx
Type I: ∫a^b ∫g1(x)^g2(x) f dy dx
Type II: ∫c^d ∫h1(y)^h2(y) f dx dy
Area: A = ■R 1 dA
POLAR
x=r cosθ, y=r sinθ, dA = r dr dθ
TRIPLE INTEGRALS
■E f dV
Mass: m = ■E ρ dV
Cylindrical: dV = r dr dθ dz
Spherical: dV = ρ^2 sinφ dρ dθ dφ
LINE INTEGRALS
r(t)=■x(t),y(t)■
ds = |r'(t)|dt
∫C f ds = ∫ f(r(t))|r'(t)|dt
Work: ∫C F·dr = ∫ F(r(t))·r'(t)dt
CONSERVATIVE FIELDS
F = ∇f
∫C F·dr = f(B)-f(A)
Test: Py = Qx
GREEN’S THEOREM
■C Pdx+Qdy = ■R (Qx - Py)dA
SURFACE AREA
A = ■ ||ru × rv|| dA