Integral Properties of Even and Odd Functions
Integrals over symmetric intervals can be made simpler by using the distinct integral features of even and odd functions. These are as follows:
Even Functions
When f(x) is an even function, its integral over the symmetric interval [−a, a] can be reduced as follows:
∫a-a f(x) dx = 2 ∫a0 f(x) dx
Due to its symmetry about the y-axis, the graph of an even function has this property. Integrating over a symmetric interval effectively doubles the area under the curve on one side, so we only need to compute half of it.
Odd Functions
When f(x) is an odd function, its integral over the symmetric interval [−a, a] can be reduced as follows:
∫a-a f(x) dx = 0
The rotational symmetry of the odd function’s graph origin gives birth to this characteristic. The net area is zero when the integration across a symmetric interval eliminates the positive and negative regions.
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Even and Odd Functions
Even and odd functions are types of functions. A function f is even if f(-x) = f(x), for all x in the domain of f. A function f is an odd function if f(-x) = -f(x) for all x in the domain of f, i.e.
- Even function: f(-x) = f(x)
- Odd function: f(-x) = -f(x)
In this article, we will discuss even and odd functions, even and odd function definitions, even and odd functions in trigonometry, and even and odd function graphs and others in detail.
Table of Content
- What are Even and Odd Functions?
- Even and Odd Functions Definition
- Even Function
- Even Function Examples
- Even and Odd Functions Graph
- Even Functions Graph
- Odd Function
- Odd Function Examples
- Odd Functions Graph
- Neither Odd Nor Even
- Even and Odd Functions in Trigonometry
- Properties of Even and Odd Functions
- Integral Properties of Even and Odd Functions
- Even and Odd Functions Examples
- Practice Questions on Even and Odd Functions
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