Riemann Limit and Fundamental Theorem of Calculus

These flashcards cover key concepts related to the Riemann limit and the Fundamental Theorem of Calculus (FTC), including definitions and formulas essential for understanding integrals.

Riemann Limit definition of the definite integral for finding the area under the curve f(x) on the interval [a, b].

∫ from a→b f(x)dx= lim n→∞ Σ (f(a+k b- a/n)b-a/n)

FTC 1

F(b)-F(a)

d/dx ∫ from g(x) to h(x) f(t)dt

f(h(x))h’(x)-F(g(x))g'(x)

Accumulation Function accumulated area under the function f starting at some constant a and ending at x.

∫ from a to x f(t) dx

Given an initial F(a) and the rate of change F’(x)=f(x)F(b)

F(b) = F(a) + ∫[a to b] f(x) dx **This is a manipulation of the FTC and often involves an integrand which must be done on the calculator

∫ from a to a f(x)dx

0

∫ from a to b f(x)dx

-∫ from b to a f(x)dx

∫ from a to b k*f(x)dx

k∫ from a to b f(x)dx

∫ from a to a (f(x)+-g(x))dx

∫ from a to b f(x)dx +- ∫ from a to b f(x)dx

∫ from a to b f(x)dx +- ∫ from b to c f(x)dx

∫ from a to c f(x)dx

FTC ll

f(x)

∫ f(x)dx

F(x)+c where F(x) is an anti derivative of f(x)

∫xⁿdx

xⁿ⁺¹/n+1 + c

∫cos udu

Sin u + c

∫sin udu

-cos u + c

∫sec² udu

tan u + c

∫csc² udu

-cot u + c

∫sec u tan udu

sec u + c

∫csc u cot udu

-csc u + c

∫e^udu

e^u + c

∫x^-1 dx or ∫dx/x

ln |x| + c

∫du/u

ln |u| + c

∫du/√a² - u²

arcsin(u/a) + c

∫du/a2 + u2

(1/a)arctan(u/a) + c

Change of variable (u substitution)

∫ from a to b f(u(x))u’(x)dx=∫ from u(a) to u(b) f(u)du

Average value of a function f(x) on the interval [a,b]

1/b-a ∫ from a to b f(x)dx

MVT for integrals: if f is continuous on [a,b] then there exists c∈[a,b] such that…

(b-a)f(c)= ∫ from a to b f(x)dx

Find the line x=c that divides the area under f(x) on [a,b] into two equal areas

∫ from a to c f(x)dx = ∫ from c to b f(x)dx or ∫ from a to b f(x)dx=2∫ from a to c f(x)dx