How is the sum calculated?

By dividing the region up into shapes

By adding the different regions

By combining all the regions together

What is the particularity of a Riemann sum?

That it is contained between the lower and upper Darboux sums

That it is contained between the lower and upper sums

That it is contained between the lower and upper Darboux functions

That it is contained between the lower and upper Darboux rules

What is the principle behind the slope of the right Riemann sum?

That "f" is approximated by the value at the right endpoint

That "R" is approximated by the value at the right endpoint

That "f(x)" is approximated by the value at the right endpoint

That "x" is approximated by the value at the right endpoint

What is the logic behind the left Riemann sum?

That it amounts to an overestimation of "f" and that it is monotonically decreasing on a given interval

That it amounts to an underestimation of "f" and that it is monotonically decreasing on a given interval

That it amounts to an overestimation of "f" and that it is monotonically increasing on a given interval

That it amounts to an overestimation of "f" and that it is monotonically constant on a given interval

Explain the principle behind the 2 dimensions rules.

It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △Ai

It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △i

It is where in 2 dimensions, each cell can be interpreted as having an area denoted by f(x)

It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △A

Accumulation And Riemann Sums Quiz

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Accumulation And Riemann Sums Quiz - Quiz

The Riemann sum can be defined as the approximation of an integral by a finite sum, with one common application being the approximation of the area of a function or line on a graph. It can also be described as the length of curves as well as other approximations. Now, complete this quiz to better test your knowledge about this application.

Questions and Answers

1.

How is the sum calculated?
- A.
  
  By dividing the region up into shapes
- B.
  
  By adding the different regions
- C.
  
  By combining all the regions together
- D.
  
  By adding the different shades
Correct Answer
A. By dividing the region up into shapes

Explanation
The sum is calculated by dividing the region up into shapes. This means that the region is divided into different geometric shapes, such as squares, triangles, or rectangles. The areas of these shapes are then calculated individually and added together to find the total sum of the region. This method allows for a more precise and accurate calculation of the sum.

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1
0
2.

How many types of Riemann sums are there?
- A.
  
  1
- B.
  
  4
- C.
  
  5
- D.
  
  2
Correct Answer
C. 5

Explanation
There are five types of Riemann sums. Riemann sums are used in calculus to approximate the area under a curve by dividing it into smaller rectangles. The five types of Riemann sums are left endpoint, right endpoint, midpoint, trapezoidal, and Simpson's rule. Each type of Riemann sum uses a different method to approximate the area under the curve, resulting in different levels of accuracy.

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3.

What is the other name for the left rule?
- A.
  
  F(x)
- B.
  
  The cumulus rule
- C.
  
  The left Riemann rule
- D.
  
  The parallel function
Correct Answer
C. The left Riemann rule

Explanation
The left Riemann rule is the other name for the left rule. This rule is a method used in calculus to approximate the area under a curve by dividing the region into rectangles and summing their areas. The left Riemann rule specifically uses the left endpoint of each rectangle to determine its height. This rule is one of the several techniques used in numerical integration to estimate definite integrals.

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4.

How is the rule that wants x*i=xi for all I called?
- A.
  
  The right Riemann shape
- B.
  
  The right Riemann side
- C.
  
  The right Riemann sum
- D.
  
  The right Riemann shade
Correct Answer
C. The right Riemann sum

Explanation
The rule that states x*i=xi for all I is called "The right Riemann sum".

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5.

What is the particularity of a Riemann sum?
- A.
  
  That it is contained between the lower and upper Darboux sums
- B.
  
  That it is contained between the lower and upper sums
- C.
  
  That it is contained between the lower and upper Darboux functions
- D.
  
  That it is contained between the lower and upper Darboux rules
Correct Answer
A. That it is contained between the lower and upper Darboux sums

Explanation
A Riemann sum is a method of approximating the area under a curve by dividing the region into smaller rectangles and summing their areas. The particularity of a Riemann sum is that it is contained between the lower and upper Darboux sums. The lower Darboux sum is the sum of the areas of the rectangles using the lowest possible height for each rectangle, while the upper Darboux sum is the sum of the areas of the rectangles using the highest possible height for each rectangle. The Riemann sum falls between these two sums, providing an approximation of the area under the curve.

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6.

How many methods of the Riemann summation exist?
- A.
  
  4
- B.
  
  5
- C.
  
  2
- D.
  
  7
Correct Answer
A. 4

Explanation
There are four methods of the Riemann summation technique. These methods include the left Riemann sum, right Riemann sum, midpoint Riemann sum, and trapezoidal rule. Each method calculates the area under a curve by dividing it into smaller rectangles or trapezoids and summing their areas. These methods are commonly used in calculus to approximate definite integrals.

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7.

What is the principle behind the slope of the right Riemann sum?
- A.
  
  That "R" is approximated by the value at the right endpoint
- B.
  
  That "f" is approximated by the value at the right endpoint
- C.
  
  That "f(x)" is approximated by the value at the right endpoint
- D.
  
  That "x" is approximated by the value at the right endpoint
Correct Answer
B. That "f" is approximated by the value at the right endpoint

Explanation
The principle behind the slope of the right Riemann sum is that "f" is approximated by the value at the right endpoint. This means that when calculating the Riemann sum, the function "f" is represented by the value of "f" at the right endpoint of each subinterval. By using this approximation, the Riemann sum can estimate the total area under the curve of the function.

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8.

What is the logic behind the left Riemann sum?
- A.
  
  That it amounts to an underestimation of "f" and that it is monotonically decreasing on a given interval
- B.
  
  That it amounts to an overestimation of "f" and that it is monotonically decreasing on a given interval
- C.
  
  That it amounts to an overestimation of "f" and that it is monotonically increasing on a given interval
- D.
  
  That it amounts to an overestimation of "f" and that it is monotonically constant on a given interval
Correct Answer
B. That it amounts to an overestimation of "f" and that it is monotonically decreasing on a given interval

Explanation
The left Riemann sum is calculated by taking the value of the function at the left endpoint of each subinterval and multiplying it by the width of the subinterval. This method of approximation tends to overestimate the area under the curve because it uses the highest value of the function within each subinterval. Additionally, since the left endpoint is used, the sum is monotonically decreasing because the function values are decreasing as we move from left to right on the interval.

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9.

What is the formula for the trapezoidal rule?
- A.
  
  A=1/2h (b1+ b2)
- B.
  
  B= 1/2h (b1+b2)
- C.
  
  A= 1/4 h (b1+b2)
- D.
  
  A= 1/2b (b+b1)
Correct Answer
A. A=1/2h (b1+ b2)

Explanation
The formula for the trapezoidal rule is A=1/2h (b1+ b2), where A represents the area of the trapezoid, h represents the height of the trapezoid, b1 represents the length of the top base, and b2 represents the length of the bottom base. This formula calculates the area of a trapezoid by taking the average of the lengths of the two bases and multiplying it by the height.

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10.

Explain the principle behind the 2 dimensions rules.
- A.
  
  It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △i
- B.
  
  It is where in 2 dimensions, each cell can be interpreted as having an area denoted by f(x)
- C.
  
  It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △A
- D.
  
  It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △Ai
Correct Answer
D. It is where in 2 dimensions, each cell can be interpreted as having an area denoted by △Ai