Evaluating Riemann Sum f(x,y) - 4x^2+y

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SUMMARY

The discussion focuses on evaluating the Riemann sum for the function f(x,y) = 4x^2 + y over the region defined by 1 ≤ x ≤ 5 and 0 ≤ y ≤ 2. The user initially calculated the sum using incorrect midpoint coordinates, resulting in a value of 168. Upon correction, the proper midpoints for the four congruent subrectangles are identified as (2, 1/2), (2, 3/2), (4, 1/2), and (4, 3/2). The integral of the function yields a value of 338.6, leading to a negative difference when comparing the Riemann sum to the integral.

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  • Understanding of Riemann sums and their application in double integrals
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  • Basic knowledge of integration techniques for functions of two variables
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Derill03
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The problem says:

evaluate 4x^2+y by breaking into four congruent subrectangles and evaluating at the midpoints, 1=<x<=5 0=<y<=2

When i setup the rectangles these are my coordinates:

(1,1/2),(1,3/2),(3,1/2),(3,3/2) and delta A = 2

My answer comes out to be 168

When i integrate the function i get 338.6

The question then asks to compute the (riemann answer - the integral) which will be negative so I am not sure if i did the riemann correctly can someone check my work and give some feedback
 
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Derill03 said:
The problem says:

evaluate 4x^2+y by breaking into four congruent subrectangles and evaluating at the midpoints, 1=<x<=5 0=<y<=2

When i setup the rectangles these are my coordinates:

(1,1/2),(1,3/2),(3,1/2),(3,3/2) and delta A = 2
This is wrong. The midpoints of the four subrectangles are at (2, 1/2), (2, 3/2), (4,1/2), and (4, 3/2). x ranges from 1 to 5, not 0 to 4. Or were you taking the left edge rather than the midpoint?

My answer comes out to be 168

When i integrate the function i get 338.6

The question then asks to compute the (riemann answer - the integral) which will be negative so I am not sure if i did the riemann correctly can someone check my work and give some feedback
There is no reason why that difference cannot be negative.
 

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