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

The integral is the limit of Riemann sums, and the Riemann sums are the areas of the rectangles with one corner at the origin and the other at the function. Those rectangles may have negative and positive areas because the function is not always above the x-axis.In summary, the problem asks to evaluate 4x^2+y by breaking it into four congruent subrectangles and evaluating at the midpoints. After setting up the rectangles, the coordinates should be (2, 1/2), (2, 3/2), (4,1/2), and (4, 3/2). The delta A is 2 and the answer comes out to be 168. Upon integrating the function
  • #1
Derill03
63
0
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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  • #2
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.
 

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

What is a Riemann Sum?

A Riemann Sum is a method of approximating the area under a curve by dividing it into smaller rectangles and summing their areas. It is an important tool in integral calculus.

How do you evaluate a Riemann Sum?

To evaluate a Riemann Sum, you first need to choose the number of rectangles to divide the curve into. Then, you calculate the width of each rectangle by dividing the total interval by the number of rectangles. Next, you plug in the x-values of the left or right endpoints of each rectangle into the given function to find the corresponding y-values. Finally, you multiply the width by the height of each rectangle and sum the results to get an approximate value of the area under the curve.

What is the purpose of evaluating a Riemann Sum?

The purpose of evaluating a Riemann Sum is to approximate the area under a curve, which can be used to solve problems in integral calculus. It is also a fundamental concept in understanding the concept of a definite integral.

How does the function f(x,y) - 4x^2+y affect the Riemann Sum?

The function f(x,y) - 4x^2+y affects the Riemann Sum by determining the height of each rectangle. The y-values of the function at each x-value will determine the height of the corresponding rectangle, thus affecting the overall value of the Riemann Sum.

What are some common applications of evaluating a Riemann Sum?

Riemann Sums are commonly used in physics, engineering, and economics to approximate the area under a curve or to solve optimization problems. They are also used in computer graphics to render images and in machine learning for data analysis and pattern recognition.

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