CALC CHP 6 TEST

LRAM

Underestimate

MRAM

RRAM

Overestimate

Riemann sum

Order of Integration

∫{b→a} f(x) dx = -∫{a→b} f(x) dx

Zero

∫{a→a} f(x) dx = 0

Constant multiple

∫{a→b} k*f(x) dx = k*∫{a→b} f(x) dx

Sum and difference

∫{a→b} (f(x) ± g(x)) dx = ∫{a→b} f(x) dx ± ∫{a→b} g(x) dx

Additivity

∫{a→b} f(x) dx + ∫{b→c} f(x) dx = ∫{a→c} f(x) dx

Max-min Inequality

If max f and min f are maximum and minimum values of f on [a,b], then:

min f⋅(b-a) ≤ ∫{a→b} f(x) dx ≤ max f⋅(b-a)

MVT for Definite Integrals

Fundamental Theorem of Calc Pt. 1

If f is continuous on [a,b], then the function

F(x) = ∫{a-x} f(t) dt

has a derivative at every point x in [a,b], and

dF/dx = d/dx ( ∫{a→x} f(t) dt ) = f(x)

Fundamental Theorem of Calc Pt. 2

If f is continuous at every point [a,b], and if F is the antiderivative of f on [a,b], then:

∫{a→b} f(x) dx = F(b) - F(a)