P1: integral(a,b) f = -integral(b,a) f (sign flip) P2: integral(a,b) f = integral(a,c) f + integral(c,b) f (additivity) P3: integral(a,b) f(x)dx = integral(a,b) f(t)dt (dummy variable) P4: integral(a,b) f(x)dx = integral(a,b) f(a+b-x)dx (King's Rule) P5: integral(0,2a) f(x)dx = integral(0,a) [f(x)+f(2a-x)]dx (Queen's Rule) P6: integral(-a,a) f = 2integral(0,a) f if even; 0 if odd P7: integral(0,nT) f = nintegral(0,T) f if f has period T P8: integral(a,a+T) f = integral(0,T) f (shift for periodic f)

Inequality Properties:

• f >= 0 on [a,b] => integral >= 0
• f >= g on [a,b] => integral f >= integral g
• m <= f <= M => m(b-a) <= integral f <= M(b-a)