April 16th, 2017, 09:55 AM  #1 
Member Joined: Feb 2017 From: East U.S. Posts: 40 Thanks: 0  Find the Riemann Sum
Find the Riemann sum for f(x) = x2 + 3x over the interval [0, 8], where x0 = 0, x1 = 1, x2 = 2, x3 = 7, and x4 = 8, and where c1 = 1, c2 = 2, c3 = 5, and c4 = 8. I'm confused. What do the c's and x's mean again? I know, I'm dumb and have no clue on where to begin. 
April 16th, 2017, 04:06 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,511 Thanks: 585 
Without further context, they could mean anything.

April 16th, 2017, 04:41 PM  #3 
Member Joined: Feb 2017 From: East U.S. Posts: 40 Thanks: 0  
April 16th, 2017, 07:22 PM  #4  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,591 Thanks: 546 Math Focus: Yet to find out.  Quote:
It's likely the x's are your known nodes. i.e., $f(x_0) = f(0) = 0$. etc.  
April 16th, 2017, 08:33 PM  #5 
Member Joined: Feb 2017 From: East U.S. Posts: 40 Thanks: 0  Well there's a graph, but I can't show you that cause it's on my ehomework. But it's literally just a graph, no other information. It looks like the (x, y) coordinates from the interval [0, 8] are (0, 0), (2, 10), (4, 25), (6, 50), (8, 85)... I don't think that's 100% accurate, but that's what I see and all the information I can get from the graph.

April 18th, 2017, 04:46 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,165 Thanks: 867 
Surely you have a textbook that deals with the 'Riemann sum"? The Riemann sum is a way to approximate the area under the graph of a function and so the integral of the function. Given f(x) and x from a to b, divide the interval, [a, b], into n subintervals. Those are the "x" values you are given: x0= 0, x1= 1, x2= 2, x3= 7, and x4= 8. Now, construct rectangles over each interval, [0, 1], [1, 2], [2, 7], and [7, 8] to approximate the area under the curve and above each interval. Take the height of each rectangle to be the value, f(x), at some point in that interval. It looks to me like those points are the "c"s you are given. That is. with c1= 1, we imagine a rectangle with base from 0 to 1 and height f(1)= 1^2+ 3(1)= 4 (notice the "^" to indicate the power "x^2" is clearer than just "x2"). That rectangle, with base 1 0= 1 and height 4 has area 4. With c2= 2, the rectangle over the interval [1, 2] has height f(2)= 2^2+ 3(2)= 10 so area (2 1)(10)= 10. With c3= 5, the rectangle over the interval [2, 7] has height f(5)= 5^2+ 3(5)= 40 so area (7 2)(40)= 200. With c4= 8, the rectangle over the interval [7, 8] has height f(8 )= 8^2+ 3(8 )= 88 so area (8 7)(88 )= 88. The Riemann sum, approximating the area under this curve, above the xaxis, between 0 and 8 is 4+ 10+ 200+ 88= 302. Last edited by Country Boy; April 18th, 2017 at 04:48 AM. 

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