### All Calculus 1 Resources

## Example Questions

### Example Question #1 : Functions

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

**Possible Answers:**

**Correct answer:**

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 #1 : How To Find Midpoint Riemann Sums

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

**Possible Answers:**

**Correct answer:**

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 #2 : How To Find Midpoint Riemann Sums

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

**Possible Answers:**

**Correct answer:**

The problem becomes this:

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

### Example Question #3 : How To Find 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.

**Possible Answers:**

**Correct answer:**

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 : How To Find Midpoint Riemann Sums

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

**Possible Answers:**

**Correct answer:**

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 #2 : Midpoint Riemann Sums

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

**Possible Answers:**

**Correct answer:**

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 #2 : How To Find 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 .

**Possible Answers:**

**Correct answer:**

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 #7 : How To Find Midpoint Riemann Sums

Solve the integral

using the midpoint Riemann sum approximation with subintervals.

**Possible Answers:**

**Correct answer:**

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 #8 : How To Find Midpoint Riemann Sums

Solve the integral

using the midpoint Riemann sum approximation with subintervals.

**Possible Answers:**

**Correct answer:**

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 : Functions

Solve the integral

using the midpoint Riemann sum approximation with subintervals.

**Possible Answers:**

1

**Correct answer:**

1

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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