To find the average of a function over a given interval of values , the most precise method is to use an integral as follows:

Now for functions that are difficult or impossible to integrate,a Riemann sum can be used to approximate the value. A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length equal to the subinterval length  , and variable heights , which depend on the function value at a given point  .

Now note that when using the method of Riemann sums to find an average value of a function, the expression changes:

We're asked to approximate the average of   over the interval 

The subintervals have length , and since we are using the midpoints of each interval, the x-values are 

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

Now, our function is unknown--we only have its derivative, -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value, .

Euler's method utilizes the following generalized formula

or for the language used in this problem

Where  is a step size. Given an intial value , it can in turn be used to approximate , where  is the number of steps taken.

To calculate this step size, consider the range of values being evaluated and the desired number of steps.

After that, it's merely a matter of taking the steps:

And from there, putting the appropriate values back into the Riemann sum approximation:

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .

Euler's method utilizes the following generalized formula

or in the case of this problem

Where  is a step size. Given an intial value , it can in turn be used to approximate , where  is the number of steps taken.

To calculate this step size, consider the range of values being evaluated and the desired number of steps.

After that, it's merely a matter of taking the steps:

And from there, putting them back into the Riemann sum approximation:

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .

Euler's method utilizes the following generalized formula

or in the case of this problem

Where  is a step size. Given an intial value , it can in turn be used to approximate , where  is the number of steps taken.

To calculate this step size, consider the range of values being evaluated and the desired number of steps.

After that, it's merely a matter of taking the steps:

And from there, putting them back into the Riemann sum approximation:

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .

Euler's method utilizes the following generalized formula

or in the case of this problem

Where  is a step size. Given an intial value , it can in turn be used to approximate , where  is the number of steps taken.

To calculate this step size, consider the range of values being evaluated and the desired number of steps.

After that, it's merely a matter of taking the steps:

And from there, putting them back into the Riemann sum approximation:

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .

Euler's method utilizes the following generalized formula

or in the case of this problem

Where  is a step size. Given an intial value , it can in turn be used to approximate , where  is the number of steps taken.

To calculate this step size, consider the range of values being evaluated and the desired number of steps.

After that, it's merely a matter of taking the steps: 

And from there, putting them back into the Riemann sum approximation:

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

Now, our function is unknown--we only have its derivative -- but we can calculate an approximation of function values using this derivative with Euler's method since we have an initial value .

Euler's method utilizes the following generalized formula

or in the case of this problem

Where  is a step size. Given an intial value , it can in turn be used to approximate , where  is the number of steps taken.

To calculate this step size, consider the range of values being evaluated and the desired number of steps.

After that, it's merely a matter of taking the steps:

And from there, putting them back into the Riemann sum approximation:

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are 

A Riemann sum integral approximation over an interval  with  subintervals follows the form:

It is essentially a sum of  rectangles each with a base of length  and variable heights , which depend on the function value at a given point  .

We're asked to approximate 

So the interval is , the subintervals have length , and since we are using the midpoints of each interval, the x-values are