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Example Questions
Example Question #1 :How To Find Midpoint Riemann Sums
Estimate the area under the curve for the following function using a midpoint Riemann sum fromtowith.
If we want to estimate the area under the curve fromtoand 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 fromto, whose height is the value of the function at, and a rectangle fromto, 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 :How To Find Midpoint Riemann Sums
Estimate the area under the curve for the following function fromtousing a midpoint Riemann sum withrectangles:
If we are told to userectangles fromto, this means we have a rectangle fromto, a rectangle fromto, a rectangle fromto, and a rectangle fromto. We can see that the width of each rectangle isbecause we have an interval that isunits long for which we are usingrectangles 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 :How To Find Midpoint Riemann Sums
Find the area underon the intervalusing five midpoint Riemann sums.
The problem becomes this:
Addings these rectangles up to approximate the area under the curve is
Example Question #4 :How To Find Midpoint Riemann Sums
Approximate the area under the curve fromusing the midpoint Riemann Sum with a partition of size five given the graph of the function.
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 #5 :How To Find Midpoint Riemann Sums
Approximate the area underneath the given curve using the Riemann Sum with eight intervals for.
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 :How To Find Midpoint Riemann Sums
Using a midpoint Reimann sum with, estimate the area under the curve fromtofor the following function:
Thus, our intervals areto,to, andto.
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 :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 ofwith, fromto.
Thus, our intervals areto,to, andto.
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 :How To Find Midpoint Riemann Sums
Solve the integral
using the midpoint Riemann sum approximation withsubintervals.
Midpoint Riemann sum approximations are solved using the formula
whereis the number of subintervals andis 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 :How To Find Midpoint Riemann Sums
Solve the integral
using the midpoint Riemann sum approximation withsubintervals.
Midpoint Riemann sum approximations are solved using the formula
whereis the number of subintervals andis 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 :How To Find Midpoint Riemann Sums
Solve the integral
using the midpoint Riemann sum approximation withsubintervals.
1
1
Midpoint Riemann sum approximations are solved using the formula
whereis the number of subintervals andis 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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