∫_{a^{b f(x)dx = F(x) [a, b] = F(b) - F(a)}}

(The integral from a to b of a function is equal to the anti-derivative of the function from a to b; this equals the function of b minus the function of a.)

d/dx ∫_{a^{x f(t)dt = f(x)}}

(The derivative of the integral from a to x of a function with respect to a variable other than x, is the function with respect to the upper x limit.)

Riemman's Sum

(Left)

delta x(y₀+y₁+...+y_n-1)

**delta x = (b-a)/n-sub intervals

Riemman's Sum

(Right)

[(delta x)/2] (1y₀+2y₁+...+1y_n)

Coef.'s

1, 2, 2, 2, 1

[(delta x)/3] (1y₀+4y₁+2y₂+...+4y_n-1+1y_n)

Coef.'s

1, 4, 2, 4, 2, 4, 1

Integration:

Indefinite Integrals

Area Under the Curve:

One Function

∫_{a^{b f(x)dx}}

Take the integral from a to b of the function.

∫_{a^{b [f(x) - g(x)]dx}}

Take the integral from a to b of the upper function minus the lower function.