Find the average value in a calculus approach

In summary, the average value of f(x) = 4sinx + 4cosx on the interval [0, 16pi/6] is 3/8pi. In order to get the correct answer, it is important to evaluate the terms correctly and factor out common factors to simplify the expression.
  • #1
silverbell
9
0

Homework Statement



Find the average value of : f(x) = 4sinx + 4cosx on the interval [0, 16pi/6]

Average value: ?

Homework Equations



Integrals

The Attempt at a Solution



1/ [(16pi/6) -0] ∫ from 0 to 16pi/6 4sinx + 4cosx dx

[3/8pi] ∫ from 0 to 16pi/6 4sinx + 4cosx dx <------simplify

[3/8pi] ∫ from 0 to 16pi/6 4sinx - 4cosx <------ integrated

[3/8pi] [4sin(16pi/6) - 4cos(16pi/6)] - [3/8pi] [ 4sin(0) - 4cos(0)] <----substitution

I get 12.296 but the answer isn't right. I don't know what I'm doing wrong. Please help me understand what I'm doing wrong. Thank you very much. :)
 
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  • #2
What did you get for 3/(8π),? (Yes, the parentheses are important.)

    "    sin(16π/6) ?

    "    cos(16π/6) ?

    "    4sin(0) - 4cos(0) ?

I would estimate that the answer is considerably smaller than 4, maybe less than 1.
 
  • #3
I got for

3/(8π) ---> 0.119

sin(16π/6) ---> 0.866

cos(16π/6) ---> -0.5

4sin(0) - 4cos(0) ---> 0 - 4
 
  • #4
Looks like you integrated it properly. I think you evaluated the terms incorrectly.

I suggest you factor the expression as much as possible to remove the 4's. That cleans up your expression and reduces chance of error when cranking through the numbers.
 
  • #5
LawrenceC said:
Looks like you integrated it properly. I think you evaluated the terms incorrectly.

I suggest you factor the expression as much as possible to remove the 4's. That cleans up your expression and reduces chance of error when cranking through the numbers.


Thanks for the tip. I got the answers. :D
 

1. What is the definition of average value in calculus?

In calculus, the average value of a function is defined as the value that the function takes on over a given interval, divided by the length of that interval.

2. How do you find the average value of a function using calculus?

To find the average value of a function using calculus, you need to first integrate the function over the given interval, and then divide the result by the length of the interval. This can be represented by the formula:

Average value = (1/b-a) * ∫ab f(x)dx

where a and b are the endpoints of the interval.

3. What is the significance of finding the average value of a function in calculus?

Finding the average value of a function in calculus is important because it allows you to determine the overall behavior of the function over a certain interval. It can also be used to find the mean value of a continuous dataset, and can be applied to real-life scenarios such as calculating average speed or rate of change.

4. What are some common methods used to find the average value of a function?

Some common methods used to find the average value of a function in calculus include the mean value theorem, the first fundamental theorem of calculus, and the midpoint rule. These methods involve using integration to find the average value of a function over a given interval.

5. Can the average value of a function be negative?

Yes, the average value of a function can be negative. This can occur when the function has negative values over the given interval, which can result in a negative average value. It is important to consider the context and meaning of the function when interpreting the sign of the average value.

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