# Calculus 1 : Functions

## Example Questions

### Example Question #41 : How To Find Midpoint Riemann Sums

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

Explanation:

A Reimann sum integral approximation of a function , with  points on the interval  follows the form:

For this problem, there will be three intervals of length  with midpoints .

Therefore, the integral approximation of is:

### Example Question #42 : How To Find Midpoint Riemann Sums

Approximate the value of the integral of the function  on the interval  using the method of Midpoint Reimann Sums and four midpoint values.

Explanation:

A Reimann sum integral approximation of a function , with  points on the interval  follows the form:

For this problem, four midpoints means four intervals of length  with midpoints

The approximation for this problem is thus:

### Example Question #43 : How To Find Midpoint Riemann Sums

Using the method of Midpoint Reimann sums, approximate the integral of  over the inteval  using two midpoints.

Explanation:

A Reimann sum integral approximation, for  points/intervals over the range , follows the form:

In the case of the problem, the intervals will have length , with midpoints .

For the function , the Reimann sum approximation is then:

### Example Question #41 : Functions

This table gives values of a function  at certain values of .

Using the table above, find the midpoint Riemann sum of  with  from  to .

Explanation:

Thus, our intervals are  to  to , and  to .

The midpoints of each interval are, respectively, , and

Next, use the data table to find the values of  at each value of :

Finally, calculate the estimation of the area using these values and

### Example Question #45 : How To Find Midpoint Riemann Sums

This table gives values of a function  at certain values of .

Using the table above, find the midpoint Riemann sum of  with  from  to .

Explanation:

Thus, our intervals are  to  and  to .

The midpoints of each interval are, respectively,  and

Next, use the data table to find the values of  at each value of :

Finally, calculate the estimation of the area using these values and :

### Example Question #46 : How To Find Midpoint Riemann Sums

Estimate the area under the curve for the function below with the midpoint Riemann sum, in the interval from  to  , using four rectangles. Round to the nearest hundredth.

Explanation:

Since we're using the midpoint sum, we need to separate our interval into 4 subintervals for each rectangle: [1,2]; [2,3]; [3,4]; and [4,5]. Each of these is 1 unit across, and the height of the rectangle is going to be the value of the function at the midpoints of these intervals.

Since the area of a rectangle is length*height, we have the following sum as our approximation:

which comes out to

We approximate this further as 6.91 and get our final answer.

### Example Question #41 : How To Find Midpoint Riemann Sums

Let .

What is the Midpoint Riemann sum on the interval  divided into four sub-intervals?

Explanation:

The interval  divided into four sub-intervals gives rectangles with vertices of the bases at

.

For the Midpoint Riemann sum, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or , , , and .

Because each sub-interval has a width of , the Midpoint Riemann sum is

### Example Question #48 : How To Find Midpoint Riemann Sums

Let .

What is the Midpoint Riemann sum on the interval  divided into four sub-intervals?

Explanation:

the interval  divided into four sub-intervals gives rectangles with vertices of the bases at

For the Midpoint Riemann sum, we need to find the rectangle heights which values come from the midpoint of the sub-intervals, or , and .

Because each sub-interval has a width of , the Midpoint Riemann sum is

### Example Question #49 : How To Find 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 #50 : How To Find Midpoint Riemann Sums

Using the method of midpoint Reimann sums, approximate the area of the region between the functions  and  over the interval  using 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 .

Furthermore, since the question is asking for the area between the two functions, it's asking for the difference between the larger function over the interval, f(x), and the smaller function, g(x).

The integral is thus: