Quick math help....

Saint Michael

Golden Member
Aug 4, 2007
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A Riemann Sum is:

(n)SIGMA(i=1) ( f( x + i * ( ( b - a ) / n ) ) * ( ( b - a ) / n ) )

Where n is the number of subintervals, b is the end of the interval, and a is the beginning of the interval.
 

Elderly Newt

Senior member
May 23, 2005
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Divide the interval into 5 equal intervals, and then find the midpoint of each interval. Using the y value of the midpoint and length of the whole interval, calculate the area. Do this for all the intervals then add their areas, and you'll have your Riemann Sum.

visual
 

Saint Michael

Golden Member
Aug 4, 2007
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Originally posted by: Eeezee
Why did you put Riemann Sum in quotes?

Because I don't really know the proper way to write out sigma notation on ATOT.

By the way OP, my Riemann Sum is for right endpoints, not midpoints. I'll leave it up to you to figure that out, which isn't too hard. Or you could just look it up in your book.
 

Saint Michael

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Aug 4, 2007
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Originally posted by: RESmonkey
No "Limit" part, correct?

And thanks! :)

You're not trying to find a definite integral, you're trying to find an approximating Riemann Sum. There is no need to consider limits, although you could write it out as a limit where n approaches 5... there would just be no point.
 

RESmonkey

Diamond Member
May 6, 2007
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It's i - .5, aka (2i -2)/2 for midpoint. It's in my notes, but I simplified the latter into i - .5 :)
 

RESmonkey

Diamond Member
May 6, 2007
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OMG I think this might be it:

10 [ 12 + 30 + 70 + 81 + 60]. Is this what they're asking for?
 

QED

Diamond Member
Dec 16, 2005
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What you are doing when forming a Reimann sum is approximating the area under a curve by breaking it up into subintervals, and then usie a rectangle (with a base length equal to the subinterval length and some height to best match the height of the curve in that subinterval). This rectangle then approximates the area under the curve in that particular subinterval. You then add up the areas of the rectangle from each subinterval to obtain your approximation.

The more subintervals you use, the smaller the base of each rectangle is, and the more accurate your approximation of the area under the curve over the whole interval is.

In your instance, you are told to use 5 fixed subintervals to evaluate the area under the curve v(t) from 0 to 50. Hence, your 5 subintervals are [0,10], [10,20], [20,30], [30,40], and [40,50]. You are further told to use the midpoint of each subinterval as the height for your rectangles.

Hence, your rectangle for the interval [0,10] has base length of 10, and a height of v(5), or 12. Hence, the area of this rectangle is 120. Using the units corresponding to your problem the base (time t) is measured in seconds, while the height (v(t)) is measure in feet per second-- so the 120 represents 120 seconds * feet/second, or 120 feet.

The rectangle for the interval [10,20] has base length of 10, and a height of v(15), or 30. Hence, the area of this rectangle is 300, which represents 300 feet.

Continue this for the other 3 intervals, add them up, and you end up with an approximation for the area under the curve v(t) from 0 to 50 (which is what the integral in Problem D represents). If you notice what the units of the area are, it should be obvious what the meaning of this integral are.
 

QED

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Dec 16, 2005
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Originally posted by: RESmonkey
OMG I think this might be it:

10 [ 12 + 30 + 70 + 81 + 60]. Is this what they're asking for?

LOL... very simply yes.

Now just figure out what the units are, and what it supposed to represent given the context of the problem!
 

Saint Michael

Golden Member
Aug 4, 2007
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Heh, it looks like my Riemann Sum wouldn't help you. I didn't even notice that the function wasn't provided, because I only read D.