Chapter 5 - Definite and Indefinite Integrals - AP Calculus AB

linearizing a function

approzimating the function for values of x close to c using the linear function

local linearity

if f is differentiable at x = c, then the linear function l(x) containing (c, f(c )) w/ slope f’(c ) is a close approximation to the graph of f for values of x close to c

dx =

change in x

dy

f’(x)dx

l(x)

f(c ) + f’(c ) (x-c)

indefinite integral

g(x) = ∫f(x)dx if and only if g’(x) = f(x)

K * f(x)dx =

K∫f(x)dx

(f(x) ± g(x))dx =

∫f(x)dx ± ∫g(x)dx + c

xⁿ dx =

1/n+1 * x^{n+1 + C}

e^xdx

e^{x + C}

b^xdx =

b^x * l/ln(b)

u-substitution

the reverse chain rule for integration, where u = a function/something inside a composition

riemann sum

Rn = ∑ f(c_k)∆x_k

definite integral notation

∫^b_{a f(x) dx}

the mean value theorem

1. f is differentiable for all values of x in the open interval (a,b) and..

2. f is continuous for all values of x in the closed interval [a,b]

then there is at least 1 number x=c in (a.b) such that

f’(c) = (f(b)-f(a))/b-a

rolles theorem

if

1. f is differentiable for all values of x in the open interval (a,b) and..

2. f is continuous for all values of x in the closed interval [a,b] and..

3. f(a) = f(b) = 0

then there is at least 1 number x=c in (a,b) such that f’(c) = 0

the fundamental theorem of calculus

if f is an integrable function and if g(x) = ∫f(x)dx then ∫^b_a f(x)dx = g(b) - g(a)

∫^b_a (f(x) ± g(x))dx =

∫^b_a f(x)dx ± ∫^b_a g(x)dx

∫^b_a K * f(x)dx =

K ∫^b_a f(x)dx

∫^b_a f(x)dx if a<c<b =

∫^c_a f(x)dx + ∫^b_cf(x)dx

∫^a_b f(x)dx =

-∫^b_a f(x)dx

area of a region between two curves

∫^b_a (f(x)-g(x)) dx

average value of a function

= 1/b-a ∫^b_a f(x)dx

volume of a solid by plane slicing

V = ∫^b_a A(x)dx

the disc method

V = ∫^b_a π f(x)²dx

washer method

V = ∫^b_a π(f(x)² - g(x)²)