Properties Of Definite Integrals & Fundamental Theorem Of Calculus

Front 1

Properties of Definite Integrals
∫ [a, a] f(x) dx =

Back 1

∫ [a, a] f(x) dx = 0

Front 2

Properties of Definite Integrals
∫ [b, a] f(x) dx =

Back 2

-∫ [a, b] f(x) dx

Front 3

Properties of Definite Integrals
∫ [a, b] ( f(x) + g(x)) dx =

Back 3

∫ [a, b] f(x) dx + ∫ [a, b] g(x) dx

Front 4

Properties of Definite Integrals
∫ [a, b] ( f(x) - g(x)) dx =

Back 4

∫ [a, b] f(x) dx - ∫ [a, b] g(x) dx

Front 5

Properties of Definite Integrals
∫ [a, b] c f(x) dx =

Back 5

c ∫ [a, b] f(x) dx

Front 6

Properties of Definite Integrals
∫ [a, c] f(x) dx + ∫ [c, b] f(x) dx =

Back 6

∫ [a, b] f(x) dx

Front 7

three Inequality Properties of Definite Integrals

Back 7

Suppose f(x) > 0 for all x in [a, b] then...
∫ [a, b] f(x) dx > 0

Suppose f(x) ≥ g(x) for all x in
[a, b] then...
∫ [a, b] f(x) dx ≥ ∫ [a, b] g(x) dx

Suppose m ≤ f(x) ≤ M for all x in [a, b] then...
m(b - a) ≤ ∫ [a, b] f(x) dx ≤ M(b - a)

(( m & M are constants representing outputs ))

Front 8

Part 1 Fundamental Theorem

Back 8

Let f(t) be continuous on interval [a, b]
Let A(x) = ∫ [a, x] f(t) dt where x is in [a, b]
then A'(x) = f(x)

Example:

F(x) = ∫ [x, π] (t + 1) dt
= - ∫ [π, x] (t + 1) dt
F'(x) = x + 1

Front 9

Part 2 Fundamental Theorem

Back 9

Let f(t) be continuous on interval
[a, b]
and let F(t) be an antiderivative for f(t), then
∫ [a, b] f(t) dt = F(b) - F(a)

Front 10

"important formula" from notes
d/dx ∫ [ g(x), q(x) ] f(t) dt =

Back 10

q'(x) f(g(x)) - g'(x) f(g(x))