# Calculus 1 : Functions

## Example Questions

### Example Question #51 : Midpoint Riemann Sums

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

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are four midpoints, the subintervals have length , and the midpoints are .

The integral is thus:

### Example Question #52 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the area between the curves  and  over the interval  using four midpoints.

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are four midpoints, the subintervals have length , and the midpoints are .

Since this problem is dealing with the area of the region between functions, it's essentially asking for the difference of the larger and smaller areas. Over the given interval, .

The integral is thus:

### Example Question #53 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the area between the curves  and  over the interval  using four midpoints.

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are four midpoints, the subintervals have length , and the midpoints are .

Since this problem is dealing with the area of the region between functions, it's essentially asking for the difference of the larger and smaller areas. Over the given interval, .

The integral is thus:

### Example Question #54 : How To Find Midpoint Riemann Sums

The velocity of an errant particle is given by the function . Approximate the average velocity of the particle over the interval of time  using the method of midpoint Reimann sums and four midpoints.

Explanation:

Average velocity can be found as total distance traveled divided by total time elapsed.

Distance traveled over an interval of time can be found by integrating the velocity function over this interval of time. For this problem, Reimann sums will be used to approximate this integral.

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are four midpoints, the subintervals have length , and the midpoints are .

The integral is thus:

Now note that this is an approximation of the distance traveled and we want the average velocity. To find this, divide this value by the time elapsed, which is :

### Example Question #55 : How To Find Midpoint Riemann Sums

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

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are five midpoints, the subintervals have length , and the midpoints are .

This gives the integral approximation:

### Example Question #56 : How To Find Midpoint Riemann Sums

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

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are four midpoints, the subintervals have length , and the midpoints are .

This gives the integral approximation:

### Example Question #57 : How To Find Midpoint Riemann Sums

Approximate the integral  using the method of midpoint Reimann sums and three midpoints.

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are three midpoints, the subintervals have length , and the midpoints are .

Thus this gives the integral approximation:

### Example Question #58 : How To Find Midpoint Riemann Sums

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

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are four midpoints, the subintervals have length , and the midpoints are .

This gives the integral approximation:

### Example Question #59 : How To Find Midpoint Riemann Sums

Approximate the integral  using the method of midpoint Reimann sums and two midpoints.

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are two midpoints, the subintervals have length , and the midpoints are .

The integral approximation becomes:

### Example Question #60 : How To Find Midpoint Riemann Sums

Approximate the integral  using the method of midpoint Reimann sums and two midpoints.

Explanation:

The Reimann sum approximation of an integral of a function with  subintervals over an interval  takes the form:

Where  is the length of the subintervals.

For this problem, since there are two midpoints, the subintervals have length , and the midpoints are .

This gives the integral approximation