So, it depends on whether the function is increasing or decreasing?

Yes. The area under the curve of a decreasing function will be overestimated by a left-hand Riemann sum and underestimated by a right-hand Riemann sum. The exact opposite is true for an increasing function.

Do the average of the three differents sums ( [L(S) + M(S) + R(S)] / 3 ) would be a better approximation? Or is it just an overkill calculation for an approximate evaluation of the area?

I would think that the M(S) would be the best approximation because it doesn't estimate too much or too less no matter what type of curve. ( [L(S) + M(S) + R(S)] / 3 ) will probably be less accurate than just M(S) because the average contains more deviations from the actual area under curve than M(S) does. I'm not too sure though. It's going to have to depend on the curve.

What if it is both increasing and decreasing? Does slope, or size of region where it is increasing or decreasing matter?

Yes, those things matter. You would have a much tougher time answering the questions above on a curve with lots of inflection points.

Main content

Course: Calculus, all content (2017 edition) > Unit 4

Lesson 8: Riemann sums

Comparing areas of Riemann sums worksheet

Google Classroom

Practice ordering the areas of a left, midpoint, and right Riemann sum from smallest to largest.

Problem 1

Let

A

denote the area of the shaded region shown below.

We can approximate the exact area

A

using the following Riemann sums.

L (6)

is the left-hand rule with

6

equal subdivisions.

M (6)

is the midpoint rule with

6

equal subdivisions.

R (6)

is the right-hand rule with

6

equal subdivisions.

Put these three approximations in order from smallest to largest.


	$L (6)$ ‍
	$M (6)$ ‍
	$R (6)$ ‍

Problem 2

Let

A

denote the area of the shaded region shown below.

We can approximate the exact area

A

using the following Riemann sums.

L (6)

is the left-hand rule with

6

equal subdivisions.

M (6)

is the midpoint rule with

6

equal subdivisions.

R (6)

is the right-hand rule with

6

equal subdivisions.

Put these three approximations in order from smallest to largest.

Graph
	$R (6)$ ‍
	$M (6)$ ‍
	$L (6)$ ‍

Problem 3

Let

A

denote the area of the shaded region shown below.

We can approximate the exact area

A

using the following Riemann sums.

L (6)

is the left-hand rule with

6

equal subdivisions.

M (6)

is the midpoint rule with

6

equal subdivisions.

R (6)

is the right-hand rule with

6

equal subdivisions.

Put these three approximations in order from smallest to largest.

Graph
	$R (6)$ ‍
	$M (6)$ ‍
	$L (6)$ ‍

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ymgli
Posted 7 years ago. Direct link to ymgli's post “So, it depends on whether...”
So, it depends on whether the function is increasing or decreasing?
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(26 votes)
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- CMcIntyre
  Posted 7 years ago. Direct link to CMcIntyre's post “Yes. The area under the c...”
  Yes. The area under the curve of a decreasing function will be overestimated by a left-hand Riemann sum and underestimated by a right-hand Riemann sum. The exact opposite is true for an increasing function.
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  (35 votes)
Gael Danguy
Posted 7 years ago. Direct link to Gael Danguy's post “Do the average of the thr...”
Do the average of the three differents sums ( [L(S) + M(S) + R(S)] / 3 ) would be a better approximation? Or is it just an overkill calculation for an approximate evaluation of the area?
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(11 votes)
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- Jason Li
  Posted 7 years ago. Direct link to Jason Li's post “I would think that the M(...”
  I would think that the M(S) would be the best approximation because it doesn't estimate too much or too less no matter what type of curve. ( [L(S) + M(S) + R(S)] / 3 ) will probably be less accurate than just M(S) because the average contains more deviations from the actual area under curve than M(S) does. I'm not too sure though. It's going to have to depend on the curve.
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  (3 votes)
Vivek Anand
Posted 7 years ago. Direct link to Vivek Anand's post “What if it is both increa...”
What if it is both increasing and decreasing?
Does slope, or size of region where it is increasing or decreasing matter?
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(5 votes)
Answer
- Ross Henderson
  Posted 7 years ago. Direct link to Ross Henderson's post “Yes, those things matter....”
  Yes, those things matter. You would have a much tougher time answering the questions above on a curve with lots of inflection points.
  Comment on Ross Henderson's post “Yes, those things matter....”
  (5 votes)
98mmeza
Posted 7 years ago. Direct link to 98mmeza's post “What happens if its incre...”
What happens if its increasing and decreasing
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(3 votes)
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