Calculus 1 : Midpoint Riemann Sums

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #171 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate the average of  over the interval  using four midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

 

Example Question #172 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate the average of  over the interval  using four midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #173 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using three midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #174 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using threemidpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #175 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using three midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #176 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using three midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #177 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using three midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #178 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using three midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #179 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using four midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

Example Question #180 : Midpoint Riemann Sums

Utilize the method of midpoint Riemann sums to approximate  using three midpoints.

Possible Answers:

Correct answer:

Explanation:

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 

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