Chapter 5 - Definite and Indefinite Integrals - AP Calculus AB

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

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

change in x

f’(x)dx

f(c ) + f’(c ) (x-c)

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

K∫f(x)dx

∫f(x)dx ± ∫g(x)dx + c

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

e^{x + C}

b^x * l/ln(b)

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

Rn = ∑ f(c_k)∆x_k

∫^b_{a f(x) dx}

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

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

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)dx ± ∫^b_a g(x)dx

K ∫^b_a f(x)dx

∫^c_a f(x)dx + ∫^b_cf(x)dx

-∫^b_a f(x)dx

∫^b_a (f(x)-g(x)) dx

= 1/b-a ∫^b_a f(x)dx

V = ∫^b_a A(x)dx

V = ∫^b_a π f(x)²dx

V = ∫^b_a π(f(x)² - g(x)²)