3: Integrals
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If we have a function f(x) which is well-defined for some a \le x \le b, its integral over those two values is defined as \int_a^b dx\; f(x) \;=\; \lim_{N \rightarrow \infty} \, \sum_{n=0}^{N} \Delta x\; f(x_n) \;\;\;\mathrm{where}\;\; x_n = a + n\Delta x, \;\; \Delta x \equiv \left(\frac{b-a}{N}\right). This is called a definite integral, and represents the area under the graph of f(x) in the region between x=a and x=b, as shown in the figure below. The function f(x) is called the integrand, and the two points a and b are called the bounds of the integral. The interval between the two bounds is divided into N segments, of length (b-a)/N each. Each term in the sum represents the area of a rectangle, and as N\rightarrow \infty, the sum converges to the area under the curve.

A multiple integral involves integration over more than one variable. For instance, when we have a function f(x_1,x_2) that depends on two independent variables, x_1 and x_2, we can perform a double integral by integrating over one variable first, then the other variable: \int_{a_1}^{b_1} dx_1 \int_{a_2}^{b_2} dx_2 \; f(x_1, x_2) \equiv \int_{a_1}^{b_1} dx_1 F(x_1)\quad\text{where}\;\;F(x_1) \equiv \int_{a_2}^{b_2} dx_2 \; f(x_1, x_2).