Riemann Sum with subintervals/partition

In summary, the conversation is about using the Riemann Sum to calculate a specific value, with a given partition and embedded points. The answer is 79. The person asking for help has tried to look for examples and videos but could not find one similar to this problem. They are seeking advice on how to approach this type of question.
  • #1
MelissaJL
50
0
So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this:
Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4].
a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P.

Now I know how to calculate other Riemann Sums but I have not encountered one with a partition and subintervals yet. I tried to do the autodidactic thing and look up examples and videos but I could not find one similar to this. If I could get some help on how I approach this type of question that would be great.

The answer is 79.

Thanks :)
 
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  • #2
MelissaJL said:
So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this:
Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4].
a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P.

Now I know how to calculate other Riemann Sums but I have not encountered one with a partition and subintervals yet. I tried to do the autodidactic thing and look up examples and videos but I could not find one similar to this. If I could get some help on how I approach this type of question that would be great.

The answer is 79.

Thanks :)

You want to compute the Riemann sum. I'll give you a hand here.. Recall that [itex]Δx_i = \frac{b-a}{n}[/itex]

What do you get when you compute this : [itex]\sum_{i=1}^{n} f(x_i)Δx_i[/itex]

Where xi is an arbitrary point in the i'th subinterval.
 
  • #3
Zondrina said:
You want to compute the Riemann sum. I'll give you a hand here.. Recall that [itex]Δx_i = \frac{b-a}{n}[/itex]

What do you get when you compute this : [itex]\sum_{i=1}^{n} f(x_i)Δx_i[/itex]

Where xi is an arbitrary point in the i'th subinterval.
The specified partition does not have equal length sub-intervals.

Δx1 = 2 = Δx3 .

Δx2 = 1 = Δx4 .
 
  • #4
MelissaJL said:
So I missed a class and am trying to figure out a question in my textbook but am completely lost. It goes a little something like this:
Let f(x)=x3 and let P=<-2,0,1,3,4> be a partition of [-2,4].
a) Compute Riemann Sum S(f,P*) if the points <x1*,x2*,x3*,x4*>=<-1,1,2,4> are embedded in P.
Okay, -1 lies in the interval [-2, 0], 1 lies in [0, 1] or [1, 3], take your pick, 2 lies in [1, 3] so we better pick [0, 1] for the 1, and 4 lies in [3, 4].
The length of [-2, 0] is 0-(-2)= 2, the length of [0, 1] is 1- 0= 1, the length of [1, 3] is 3- 1= 2, and the length of [3, 4] is 4- 3= 1.

So the Riemann sum is f(-1)(2)+ f(1)(1)+ f(2)(2)+ f(4)(1) = -13(2)+ 13(1)+ 23(2)+ 43(1)

Now, I know how to calculate other Riemann Sums but I have not encountered one with a partition and subintervals yet.
That's impossible- every Riemann Sum requires a partition. And, it happens, there are NO "subintervals" given unless you mean that the partition itself breaks the interval [-1, 4] into "subintervals". But, again, that is true of every Riemann Sum.

I tried to do the autodidactic thing and look up examples and videos but I could not find one similar to this. If I could get some help on how I approach this type of question that would be great.

The answer is 79.

Thanks :)
 

1. What is a Riemann Sum with subintervals/partition?

A Riemann Sum with subintervals/partition is a method used in calculus to approximate the area under a curve by dividing the interval into smaller subintervals and calculating the sum of the areas of rectangles within those subintervals.

2. How is a Riemann Sum with subintervals/partition calculated?

To calculate a Riemann Sum with subintervals/partition, you need to first determine the width of each subinterval by dividing the total interval by the number of subintervals. Then, evaluate the function at each subinterval and multiply it by the width of the subinterval. Finally, add all of these values together to get the approximate area under the curve.

3. What is the purpose of using subintervals/partition in a Riemann Sum?

By using subintervals/partition, we can break a complex shape into smaller, simpler shapes that are easier to calculate. This allows us to get a more accurate approximation of the area under a curve.

4. How do you choose the number of subintervals for a Riemann Sum?

The number of subintervals for a Riemann Sum is typically chosen based on the level of accuracy needed. The more subintervals used, the closer the approximation will be to the actual area under the curve.

5. Can a Riemann Sum with subintervals be used for non-linear functions?

Yes, a Riemann Sum with subintervals can be used for both linear and non-linear functions. However, the calculations may become more complex for non-linear functions, and more subintervals may be needed for a more accurate approximation.

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