# Calculus 1 : Differential Functions

## Example Questions

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### Example Question #1 : Midpoint Riemann Sums

Estimate the area under the curve for the following function using a midpoint Riemann sum from    to    with  .

Explanation:

If we want to estimate the area under the curve from    to    and are told to use  ,  this means we estimate the area using two rectangles that will each be two units wide and whose height is the value of the function at the midpoint of the interval. We have a rectangle from    to  ,  whose height is the value of the function at  ,  and a rectangle from    to  ,  whose height is the value of the function at  .  First we can find the value of the function at these midpoints, and then add the areas of the two rectangles, which gives us the following:

### Example Question #2 : Midpoint Riemann Sums

Estimate the area under the curve for the following function from    to    using a midpoint Riemann sum with    rectangles:

Explanation:

If we are told to use   rectangles from    to  ,  this means we have a rectangle from    to  ,  a rectangle from    to  ,  a rectangle from    to  ,  and a rectangle from    to  .  We can see that the width of each rectangle is    because we have an interval that is    units long for which we are using    rectangles to estimate the area under the curve. The height of each rectangle is the value of the function at the midpoint for its interval, so first we find the height of each rectangle and then add together their areas to find our answer:

### Example Question #3 : Midpoint Riemann Sums

Find the area under  on the interval  using five midpoint Riemann sums.

Explanation:

The problem becomes this:

Addings these rectangles up to approximate the area under the curve is

### Example Question #4 : Midpoint Riemann Sums

Approximate the area under the curve from  using the midpoint Riemann Sum with a partition of size five given the graph of the function.

Explanation:

We begin by finding the given change in x:

We then define our partition intervals:

We then choose the midpoint in each interval:

Then we find the value of the function at the point.  This is determined through observation of the graph

Then we simply substitute these values into the formula for the Riemann Sum

### Example Question #1 : Midpoint Riemann Sums

Approximate the area underneath the given curve using the Riemann Sum with eight intervals for .

Explanation:

We begin by defining the size of our partitions and the partitions themselves.

We then choose the midpoint in each interval:

Then we find the function value at each point.

We then substitute these values into the Riemann Sum formula.

### Example Question #6 : Midpoint Riemann Sums

Using a midpoint Reimann sum with , estimate the area under the curve from  to  for the following function:

Explanation:

Thus, our intervals are  to  to , and  to .

The midpoints of each interval are, respectively, , and .

Next, we evaluate the function at each midpoint.

Finally, we calculate the estimated area using these values and

### Example Question #7 : Midpoint Riemann Sums

The table above gives the values for a function at certain points.

Using the data from the table, 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 take the values the function at each midpoint.

Finally, we calculate the estimated area using these values and .

### Example Question #8 : Midpoint Riemann Sums

Solve the integral

using the midpoint Riemann sum approximation with  subintervals.

Explanation:

Midpoint Riemann sum approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The approximate value at each midpoint is below.

The sum of all the approximate midpoints values is , therefore

### Example Question #9 : Midpoint Riemann Sums

Solve the integral

using the midpoint Riemann sum approximation with  subintervals.

Explanation:

Midpoint Riemann sum approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The approximate value at each midpoint is below.

The sum of all the approximate midpoints values is , therefore

### Example Question #10 : Midpoint Riemann Sums

Solve the integral

using the midpoint Riemann sum approximation with  subintervals.

1

1

Explanation:

Midpoint Riemann sum approximations are solved using the formula

where  is the number of subintervals and  is the function evaluated at the midpoint.

For this problem, .

The approximate value at each midpoint is below.

The sum of all the approximate midpoints values is , therefore

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