Average value of a curve

In summary, the average value of a function over an interval is given by the formula f_{ave}=\frac{1}{b-a}\int_{x=a}^{b}f(x)dx, where f(x) is properly integrable over the interval [a,b]. For a constant function, the area under the curve from a to b is given by c(b-a), while for a variable function it is represented by \int_a^b f(x)dx. In the case of a sinusoid, which is an odd function, the average value over a symmetric interval around 0 will be 0.
  • #1
perryben
8
0
If I had a sinusoid, how would I find the average value of it over a given interval. Say -pi/5 to pi/5 for instance. Thanks everybody.
 
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  • #2
The average value of a function, say [tex]f(x),[/tex] over the interval [a,b] is given by the the formula

[tex]f_{\mbox{ave}}=\frac{1}{b-a}\int_{x=a}^{b}f(x)dx[/tex]

where I have assumed that [tex]f(x),[/tex] is properly integrable over [a,b].
 
  • #3
The point is: if you had a constant function, f(x)= c, the "area under the curve" from a to b would f(x)(b-a)= c(b-a). With a variable function, that area is [tex]\int_a^b f(x)dx[/tex].

If fave is the average of the function we must have
[tex]\int_a^b f(x)dx= f_{ave}(b-a)[/tex]
 
  • #4
If I had a sinusoid, how would I find the average value of it over a given interval. Say -pi/5 to pi/5 for instance. Thanks everybody.
The sine function is odd. Therefore the average over an interval symmetric around 0 will be 0.
 

1. What is the average value of a curve?

The average value of a curve, also known as the mean value, is the measure of central tendency that represents the middle value of a set of data points. It is calculated by adding all the data points together and dividing by the total number of data points.

2. How is the average value of a curve calculated?

To calculate the average value of a curve, you need to first find the total area under the curve using integration. Then, divide this area by the length of the curve to get the average value.

3. Can the average value of a curve be negative?

Yes, the average value of a curve can be negative. This occurs when the curve has both positive and negative values that cancel each other out, resulting in a negative average value.

4. What does the average value of a curve represent?

The average value of a curve represents the point on the x-axis where the curve is balanced. It is the point where the area under the curve on one side is equal to the area under the curve on the other side.

5. Why is the average value of a curve important?

The average value of a curve is important because it provides a single value that summarizes the entire curve. It is also used in various applications, such as calculating the average rate of change, finding the average velocity, and determining the average value of a function over a given interval.

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