notes/education/math/MATH1220 (calc II)/Integral Review.md

Formal Definition

Let f be a continuous function on an interval [a, b].

Divide [a, b] into n equal parts of width \Delta x = \dfrac{b-a}{n}.

Let x_0, x_1, x_2, \cdots, x_n be the endpoints of this subdivision. x_0 = a and x_n = b.

Define \int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^nf(x_i)\Delta x

• \Delta x refers to the width of each sub-interval
• f(x_i) refers to the height of each subinterval, and can be found with the equation x_i = \Delta xi + a

Then let f be a continous function on [a, b] and let F be the antiderivative of f (i.e F'(x) = f(x)). Then \int_a^b f(x) dx = F(b) - F(a).

Examples

\int_0^1 x^2 dx = \frac{1}{3} x^3 \Big |_0^1 = 1/3(1^3)- 1/3 (0)^3 = 1/3