# Calculus 1 : Functions

## Example Questions

### Example Question #91 : Midpoint Riemann Sums

Using the method of midpoint Riemann sums, approximate the integral  using three midpoints.

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 are approximating

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

### Example Question #92 : Midpoint Riemann Sums

Approximate the integral  using the method of midpoint Riemann sums and four midpoints.

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 are approximating

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

### Example Question #93 : Midpoint Riemann Sums

Using the method of midpoint Riemann sums, approximate the integral  using four midpoints.

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 approximating

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

### Example Question #94 : Midpoint Riemann Sums

Using the method of midpoint of Reimann sums, approximate the integral  using three midpoints.

Explanation:

A Reimann 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 approximating

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

### Example Question #95 : Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using four midpoints.

Explanation:

A Reimann 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 approximating

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

### Example Question #91 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using two midpoints.

Explanation:

A Reimann 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 approximating

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

### Example Question #97 : Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using three midpoints.

Explanation:

A Reimann 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 approximating

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

### Example Question #98 : Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using three midpoints.

Explanation:

A Reimann 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 approximating

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

### Example Question #99 : Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using four midpoints.

Explanation:

A Reimann 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 approximating

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

### Example Question #100 : Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the integral  using three midpoints.