$$\require{cancel}$$

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

3: Integrals is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Y. D. Chong via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.