How to Calculate the Average Value of a Sinusoid Over a Given Interval?

Click For Summary
To calculate the average value of a sinusoid over a given interval, the formula f_ave = (1/(b-a)) * ∫[a to b] f(x) dx is used, where f(x) is the sinusoidal function. For an interval such as -π/5 to π/5, the sine function, being odd, results in an average value of 0 due to symmetry. The integral of an odd function over a symmetric interval around zero yields zero, confirming that the average value is indeed 0. This principle applies to any sinusoidal function over symmetric intervals. Understanding these properties is crucial for accurately calculating averages of trigonometric functions.
perryben
Messages
8
Reaction score
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.
 
Physics news on Phys.org
The average value of a function, say f(x), over the interval [a,b] is given by the the formula

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

where I have assumed that f(x), is properly integrable over [a,b].
 
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 \int_a^b f(x)dx.

If fave is the average of the function we must have
\int_a^b f(x)dx= f_{ave}(b-a)
 
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.
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
View full post »

Similar threads

Undergrad What is the average value of a bounded periodic function over a period?
  • · Replies 3 ·
Replies
3
Views
2K
MHB 5.5.2 average value of sqrt{x} [0,4]
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Limits confusion -- What is meant by "instantaneous rate of change"?
  • · Replies 18 ·
Replies
18
Views
2K
MHB Find the average or mean slope of the function on a interval
  • · Replies 1 ·
Replies
1
Views
2K
Formula for the average EMF of generator
  • · Replies 6 ·
Replies
6
Views
931
MHB What is the Derivation of the Average Value of a Waveform Using Calculus?
  • · Replies 5 ·
Replies
5
Views
3K
MHB Solving problems using the Average Value function
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Averages and average speed/instantaneous speed relation
  • · Replies 10 ·
Replies
10
Views
2K
Exploring the Physical Application of Calculating Average Value of a Function
  • · Replies 10 ·
Replies
10
Views
3K
Average velocity or average values of functions and calculus
  • · Replies 2 ·
Replies
2
Views
5K