3: Integrals
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: \[\int_{a_1}^{b_1} dx_1 \int_{a_2}^{b_2} dx_2 \; f(x_1, x_2) \equiv \int_{a_1}^{b_1} dx_1 F(x_1)\quad\text{where}\;\;F(x_1) \equiv \int_{a_2}^{b_2} dx_2 \; f(x_1, x_2).\]