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3: Integrals

Last updated

Apr 30, 2021
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- 2.5: Exercises
- 3.1: Basic Properties of Definite Integrals

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If we have a function which is well-defined for some , 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 that depends on two independent variables, and , we can perform a double integral by integrating over one variable first, then the other variable: $\int_{a_1}^{b_1} dx_1 \int_{a_2}^{b_2} dx_2 \; f(x_1, x_2) \equiv \int_{a_1}^{b_1} dx_1 F(x_1)\quad\text{where}\;\;F(x_1) \equiv \int_{a_2}^{b_2} dx_2 \; f(x_1, x_2).$

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