# Calculus 1 : Functions

## Example Questions

### Example Question #31 : How To Find Midpoint Riemann Sums

Approximate the integral of the function  for  to  using midpoint Reimann sums and two midpoints.

Explanation:

The form of a Riemann sum follows:

Since two mid points are asked for, divide the interval into two: , and find the mid points of those: .

Therefore, the approximation of the integral is:

### Example Question #31 : Midpoint Riemann Sums

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

Explanation:

The form of a Riemann sum follows:

For two intervals of length , the midpoints are .

Therefore, the approximation of the integral is:

### Example Question #33 : How To Find Midpoint Riemann Sums

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

Explanation:

The form of a Riemann sum follows:

The interval  can be divided into three intervals of length

, with midpoints of .

Therefore, the approximation of the integral is:

### Example Question #34 : How To Find Midpoint Riemann Sums

Using midpoint Riemann sums and three mid points, approximate the area between the functions  and .

Explanation:

The form of a Riemann sum follows:

It can be applied to combinations of functions, i.e.

To find the interval for this problem, find the intersections of the two functions, i.e. the values of  that satisfy

Knowing this interval, it can be divided into three intervals of length :

, with midpoints .

So the approximation of the area between functions is:

### Example Question #35 : How To Find Midpoint Riemann Sums

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

Explanation:

The form of a Riemann sum follows:

The interval  can be divided into four intervals of length

.

With the midpoints

.

Therefore, the approximation of the integral is:

### Example Question #36 : How To Find Midpoint Riemann Sums

Find the midpoint Reimann sum for the integral of the function  over the interval  using four midpoints.

Explanation:

The form of a Riemann sum follows:

The interval  can be divided into four intervals of length

with mid points .

The Riemann sum is then

### Example Question #37 : How To Find Midpoint Riemann Sums

Use midpoint Riemann Sums to approximate the area between the -axis and the function for . Use seven intervals .

Explanation:

First, graph the equation to see if it is above the x-axis - positive, or below - negative. Then we can visualize the intervals and their midpoints:

Dividing the region from to into 7 intervals gives each a length of exactly 1.

We will be calculating the sum where is the nth interval's midpoint. Since is jus 1, in this case we're just adding together this sum:

Evaluating at these points and adding the results together yields

### Example Question #38 : How To Find Midpoint Riemann Sums

Use Riemann midpoint sums to approximate the area between the curve and the -axis between and . Use five intervals .

Explanation:

We are looking to approximate the area between 0 and 3, and we are dividing it into 5 intervals, so each interval will have a length of .

The midpoints of the 5 intervals are at , so we'll be evaluating the function for these values of x.

The formula for the Riemann midpoint sum is .

Note that you get the same answer regardless if you multiply each individual times or calculate the sum first then multiply by 0.6.

In this case, adding together the heights at the midpoints and multiplying by 0.6 yields

### Example Question #39 : How To Find Midpoint Riemann Sums

Consider the function  between  and . Find the definite integral using midpoint Riemann sums with two rectangles.

Explanation:

Midpoint sums are given by the formula

,

where n is the number of steps and the x values are the midpoints of the rectangles.

Since we have 2 steps, there would be 2 rectangles, from  and the other one . The heights of these rectangles are calculated at the midpoint of the two ends, which occur at  and

In our case, the rectangle midpoints are at 1.25 and 1.75, since we only have 2 steps.

Therefore, the value is

### Example Question #40 : How To Find Midpoint Riemann Sums

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