# Midpoint Rule

#### Squish

##### New member
I've been trying to solve this question for an good amount of time with no avail, so I've come to seek help on this one:

The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 AM by the US Army Corps of Engineers at a lake in Georgia. Use the Midpoint Rule to estimate the amount of water that flowed into this lake from July 18th, 2013 at 7:30 AM to July 26th at 7:30 AM.

I've tried evaluating the midpoints like this: (y1+y2)/2 , (y2+y3)/2 , etc. , summing them all and then multiplying them by 60*60*24 (seconds per day). That was incorrect.
I tried using the sum of y2, y4, y6, y8 and then multiplying that by 2*(seconds per day). That also didn't work, so I am currently stumped on this last question on my assignment.

#### tkhunny

##### Moderator
Staff member
I've been trying to solve this question for an good amount of time with no avail, so I've come to seek help on this one:

The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 AM by the US Army Corps of Engineers at a lake in Georgia. Use the Midpoint Rule to estimate the amount of water that flowed into this lake from July 18th, 2013 at 7:30 AM to July 26th at 7:30 AM.

View attachment 11064
I've tried evaluating the midpoints like this: (y1+y2)/2 , (y2+y3)/2 , etc. , summing them all and then multiplying them by 60*60*24 (seconds per day). That was incorrect.
I tried using the sum of y2, y4, y6, y8 and then multiplying that by 2*(seconds per day). That also didn't work, so I am currently stumped on this last question on my assignment.
Please show your calculations and explain why you believe the result is incorrect.

#### Squish

##### New member
Please show your calculations and explain why you believe the result is incorrect.
For the first attempt, I got the sum of the midpoints as such: (5257+6385)/2 + (6385+2535)/2 + (2535+4257)/2 + (4257+3011)/2 + (3011+3837)/2 + (3837+2468)/2 + (2468+3647)/2 + (3647+2996)/2 = 30266.5

then multiplied by the seconds per day: 86400*30266.5 = 2615025600 ft^3

In my second attempt I used the values {6385, 4257, 3837, 3647}, summed them up and multiplied them by 2*(seconds per day): (2*86400)*(18126) = 3132172800 ft^3

Both answers were marked incorrect on webassign.net, my assignment source.

#### Dr.Peterson

##### Elite Member
I've been trying to solve this question for an good amount of time with no avail, so I've come to seek help on this one:

The table shows the rate of inflow of water, in cubic feet per second, as measured every morning at 7:30 AM by the US Army Corps of Engineers at a lake in Georgia. Use the Midpoint Rule to estimate the amount of water that flowed into this lake from July 18th, 2013 at 7:30 AM to July 26th at 7:30 AM.

View attachment 11064
I've tried evaluating the midpoints like this: (y1+y2)/2 , (y2+y3)/2 , etc. , summing them all and then multiplying them by 60*60*24 (seconds per day). That was incorrect.
I tried using the sum of y2, y4, y6, y8 and then multiplying that by 2*(seconds per day). That also didn't work, so I am currently stumped on this last question on my assignment.
What you are doing appears to be a version of the trapezoidal approximation (averaging y values), not the midpoint rule (averaging x values). Your second attempt may be the midpoint rule. Both are explained here: http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx.

As these are different approximations, they should give answers that are reasonably close, but different. (The fluctuations in the data are such that the midpoint rule will not be very accurate.) Are you saying that you were given the correct answer and yours is very different, or only slightly different? We'll need to see both your answer and the given answer, to see which is wrong according to the instructions. Also please show us what you actually did, not just a description of it, so we can be sure whether you did each method correctly.

#### Squish

##### New member
What you are doing appears to be a version of the trapezoidal approximation (averaging y values), not the midpoint rule (averaging x values). Your second attempt may be the midpoint rule. Both are explained here: http://tutorial.math.lamar.edu/Classes/CalcII/ApproximatingDefIntegrals.aspx.

As these are different approximations, they should give answers that are reasonably close, but different. (The fluctuations in the data are such that the midpoint rule will not be very accurate.) Are you saying that you were given the correct answer and yours is very different, or only slightly different? We'll need to see both your answer and the given answer, to see which is wrong according to the instructions. Also please show us what you actually did, not just a description of it, so we can be sure whether you did each method correctly.
It doesn't show the answer, but it only marks it as either correct or incorrect on webassign.net. Also, with all due respect, I've already written my steps and answers in the reply just above yours, so I don't think I should rewrite it.

Staff member

#### Dr.Peterson

##### Elite Member
It doesn't show the answer, but it only marks it as either correct or incorrect on webassign.net. Also, with all due respect, I've already written my steps and answers in the reply just above yours, so I don't think I should rewrite it.
Because of moderation, your reply was not visible when I wrote. No, of course you don't have to rewrite it; I just wish the system could call my attention to an answer-before-the-fact like this.

The second attempt looks right to me; possibly they told you how to round (it should really have only four significant digits), and marked it wrong for that without telling you that you were close.

Or, since they gave you a lot of data that you didn't use with this method, possibly they taught (or expect you to have been taught) a modified version that uses overlapping segments. I don't suppose they provide hints or examples to show you what they are looking for, as some systems do?

You'll have to ask your instructor, in any case. (Some systems provide a way to do that in connection with a specific problem; I haven't used webassign.)

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