How do you find the Riemann sum of this function? How do I go about figuring this out? Here's the full question: You are given a function f, an interval, partition points that define a partition...

How do you find the Riemann sum of this function? How do I go about figuring this out?

Here's the full question: 

You are given a function f, an interval, partition points that define a partition P, and points xi* in the ith subinterval. Find the Riemann sum.

f(x) = 7 - 2x, [1, 5], {1, 1.6, 2.2, 3.0, 4.2, 5}, xi* = midpoint 

 I've done RAM before, but this seems nothing like it. Thanks in advance!

Asked on by modesthief

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embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

The Riemann sum is `sum_(i=1)^n f(c_i)Deltax_i` where `c_i` is a point in the ith subinterval and `Deltax_i` is the width of the ith subinterval.

We are given the function `f(x)=7-2x` on the interval [1,5].

The endpoints of the subintervals are given, and we are asked to use the midpoints of the subintervals for `c_i` .

For [1,1.6] we have `c_1=1.3` and `Deltax_1=.6`

For [1.6,2.2] we have `c_2=1.9` and `Deltax_2=.6`

For [2.2,3.0] we have `c_3=2.6` and `Deltax_3=.8`

For [3.0,4.2] we have `c_4=3.6` and `Deltax_4=1.2`

For [4.2,5] we have `c_5=4.6` and `Deltax_5=.4`

Then `sum_(i=1)^5f(c_i)Deltax_i`

`=f(1.3).6+f(1.9).6+f(2.6).8+f(3.6)1.2+f(4.6).4`

`=(4.4).6+3.2(.6)+1.8(.8)+(-.2)(1.2)+(-2.2)(.4)`

`=4.88`

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The Riemann sum using the given subintervals and points in the subintervals on the given interval is 4.88

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