# Calculus 2 : Average Values and Lengths of Functions

## Example Questions

### Example Question #11 : Average Values And Lengths Of Functions

What is the average value of the function  on the interval ?

Explanation:

The average value of a function is found by taking the integral of the function over the interval and dividing by the length of the interval.

The general rule for this type of integration is as follows.

So we have:

### Example Question #11 : Average Values And Lengths Of Functions

What is the average value of the function  on the interval ?

Explanation:

The average value of a function is found by taking the integral of the function over the interval and dividing by the length of the interval.

The general rule for this type of integration is as follows.

So we have:

### Example Question #12 : Average Values And Lengths Of Functions

Find the length L of the graph of the function f(x) on the given interval.

Explanation:

To find the length of a line along a given function f(x) from a to b, utilize the equation

Applying the equation for this problem, we first find the first derivative of the function:

Then, we use this expression in the above formula:

### Example Question #14 : Average Values And Lengths Of Functions

Let f(x) bet a function continuous on the defined below and having the derivative

Find the length L of the graph of f(x) on the interval

Explanation:

To find the length of a line along a given function f(x) from a to b, utilize the equation

Applying the equation for this problem and using the given expression for f'(x):

### Example Question #13 : Average Values And Lengths Of Functions

Find the average value of the function over the interval .

Explanation:

To find the average value of the function, we use

Average value ,

with in this case.

So we have

Using integration by parts with and hence , we have

### Example Question #16 : Average Values And Lengths Of Functions

Find the arc length of the graph on the interval .

Explanation:

The arc length for a function on the interval is given by the following integral:

.

Thus we must compute the value of for the function .

To do so, we use the chain rule:

.

Now plugging to the arc length formula, and using the interval , we obtain,

.

Using the trig identity , our integral becomes:

.

In the last step above, we invoked the property

.

Since the expression in the absolute value is positive, we can get rid of the absolute value brackets.

Thus we obtain

.

### Example Question #17 : Average Values And Lengths Of Functions

What is the average value of  on the interval ?

Explanation:

To find the average value of a function, you must use the following equation:

.  Now, we simply substitute our values, and integrate.

.  Now, we plug in our bounds (upper - lower).

### Example Question #14 : Average Values And Lengths Of Functions

Find the average value of the function  from  to

Explanation:

The average value equation is

Now, all we have to do is plug in our limits.

The second term drops out because the natural log of  equals zero.  In the first term, natural log and  cancel each other out, just leaving the  out front.

### Example Question #11 : Average Values And Lengths Of Functions

Determine the average value of the function

On the interval

Explanation:

The average value of a function on the interval  is defined as

As such, we solve the expression

As such, the average value is

### Example Question #161 : Integral Applications

Determine the average value of the function

On the interval