Consider an n x p matrix A and a p xm matrix B such that ker(A) = {0} and ker(Z) = {0}

MATH 2450 WEEK 8 Ch 12 Double Integral in Rectangular Region ʃʃRf(x,y) dA R: a ≤ x ≤ b and c ≤ y ≤ d 2D: XY plane 3D: XYZ dimension, Projection on the XY plane Integral 2D b u ʃaf(x) dx = lim u ->∞∑ i =1f(i) ∆x Linearity Rule ʃʃR(c1*f(x,y) + 2 *g(x,y)) dA c1* ʃʃRf(x,y) dA + c 2ʃʃRg(x,y) dA Dominance Rule When f(x,y) ≥ g(x,y) for all (x,y) in R Then ʃʃ R(x,y) dA ≥ ʃʃ R(x,y) dA Subdivision Rule R = R 1 R 2 ʃʃRf(x,y) dA = ʃʃR1 f(x,y) dA + ʃʃR2 f(x,y) dA if f(x,y) ≥ 0 for all (x,y) in R ʃR f(x,y) dA = volume above where R is in between z = f(x,y) and z =0 EX. ʃʃR(2-y) dA R is in the rectangle with vertices of (0,0) , (3,0) , (3,2) , (0,2) Note: the equation z = 2-y