Riemann Sum: Meaning, Formula, Limit & Method | StudySmarter

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Riemann Sum

Riemann Sum

Finding the area below a curve is not straightforward. You can take a look at our Approximating Areas article and see the different ways of using rectangles to approximate the area below a curve.

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All Riemann sums are either a left-endpoint approximation or a right-endpoint approximation.

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The right-endpoint approximation is a particular case of a Riemann sum.

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An upper Riemann sum is always greater than or equal to the exact area below the curve.

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A lower Riemann sum is always less than or equal to the exact area below the curve.

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Suppose that, in some interval, a lower Riemann sum of a function is 24.92 and an upper Riemann sum is 25.08. What can we conclude from this information?

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A left-endpoint approximation always gives us a lower Riemann sum.

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All Riemann sums are either a left-endpoint approximation or a right-endpoint approximation.

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The right-endpoint approximation is a particular case of a Riemann sum.

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An upper Riemann sum is always greater than or equal to the exact area below the curve.

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A lower Riemann sum is always less than or equal to the exact area below the curve.

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Suppose that, in some interval, a lower Riemann sum of a function is 24.92 and an upper Riemann sum is 25.08. What can we conclude from this information?

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A left-endpoint approximation always gives us a lower Riemann sum.

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Last Updated: 22.07.2022
Published at: 30.06.2022
11 min reading time

Applied Mathematics
Calculus

Absolute Maxima and Minima
Absolute and Conditional Convergence
Accumulation Function
Accumulation Problems
Algebraic Functions
Alternating Series
Antiderivatives
Application of Derivatives
Application of Higher Order Derivatives
Application of Integrals in Biology and Social Sciences
Applications of Continuity
Applications of Double Integrals
Approximating Areas
Arc Length of a Curve
Area Between Two Curves
Arithmetic Series
Arithmetic of Complex Numbers
Average Rate of Change of Populations
Average Value Function
Average Value of a Function
Bifurcation Theory
Boundary Value Problems
Calculus Linear Approximation
Calculus Of Parametric Curves
Candidate Test
Change of Variables in Multiple Integrals
Combining Different Rules
Combining Differentiation Rules
Combining Functions
Complex Analysis
Concavity of a Function
Continuity
Continuity Equations
Continuity Over an Interval
Continuity and Indeterminate Forms
Convergence Tests
Cost and Revenue
Critical points
Curl and Divergence
Curve Sketching Techniques
Density and Center of Mass
Derivative Functions
Derivative Properties
Derivative as a Limit
Derivative of Exponential Function
Derivative of Inverse Function
Derivative of Logarithmic Functions
Derivative of Trigonometric Functions
Derivative of Vector Valued Function
Derivatives
Derivatives and Continuity
Derivatives and the Shape of a Graph
Derivatives of Exponential Functions
Derivatives of Inverse Trigonometric Functions
Derivatives of Logarithmic Functions
Derivatives of Polar Functions
Derivatives of Sec, Csc and Cot
Derivatives of Sin, Cos and Tan
Determining Volumes by Slicing
Difference Quotient
Differentiability
Differentials
Differentiation of Functions of Several Variables
Direction Fields
Disk Method
Divergence Test
Double Integral
Double Integrals Over General Regions
Double Integrals Over Rectangular Regions
Double Integrals in Polar Coordinates
Dynamical Systems
Eliminating The Parameter
Euler's Method
Evaluating a Definite Integral
Evaluation Theorem
Exact equations
Exponential Functions
Exponential Model
Extrema
Finding Limits
Finding Limits of Specific Functions
First Derivative Test
Function Transformations
Fundamental Theorem of Line Integrals
Galois Theory
General Solution of Differential Equation
Geometric Series
Gradient Descent
Graphing and Optimization
Green's Function
Green's Theorem
Growth Rate of Functions
Harmonic Functions
Higher Order Partial Derivatives
Higher-Order Derivatives
Hydrostatic Pressure
Hyperbolic Functions
Implicit Differentiation Tangent Line
Implicit Relations
Improper Integrals
Increasing and Decreasing Functions
Indefinite Integral
Indeterminate Forms
Indeterminate Forms of Limits
Initial Value Problem Differential Equations
Integral Equations
Integral Test
Integrals in Economics
Integrals of Exponential Functions
Integrals of Motion
Integrating Even and Odd Functions
Integration Formula
Integration Tables
Integration Techniques
Integration Using Long Division
Integration Using Tables
Integration fundamentals
Integration of Logarithmic Functions
Integration of Vector Valued Functions
Integration using Inverse Trigonometric Functions
Intermediate Value Theorem
Inverse Trigonometric Functions
Jump Discontinuity
Lagrange Error Bound
Lagrange Multiplier
Laplace Transform
Lebesgue Integration
Limit Applications
Limit Laws
Limit of Vector Valued Function
Limit of a Sequence
Limits
Limits and Continuity
Limits at Infinity
Limits at Infinity and Asymptotes
Limits of a Function
Line Integrals
Linear Approximations and Differentials
Linear Differential Equation
Linear Functions
Logarithmic Differentiation
Logarithmic Functions
Logistic Differential Equation
Logistic Model
Maclaurin Series
Manipulating Functions
Matrix Theory
Maxima and Minima
Maxima and Minima Problems
Mean Value Theorem for Integrals
Measure Theory
Michaelis Menten Equation
Models for Population Growth
Motion Along A Line
Motion In Space
Multiple Integrals
Multivariable Calculus
Natural Logarithmic Function
Net Change Theorem
Newton's Method
Non Differentiable Functions
Nonhomogeneous Differential Equation
Nonlinear Differential Equations
Numerical Linear Algebra
One-Sided Limits
Optimization Problems
Optimization Problems in Economics
Optimization Theory
Ordinary Differential Equations
P Series
PDE Solutions
Parametric Surface Area
Parametric derivatives
Partial Derivatives
Particle Model Motion
Particular Solutions to Differential Equations
Piecewise Defined Function
Polar Coordinates
Polar Coordinates Functions
Polar Curves
Population Change
Power Series
Probability Theory
Properties of Definite Integrals
Radius of Convergence
Ratio Test
Real Analysis
Related Rates
Removable Discontinuity
Revenue as Average Rate of Change
Riemann Integral
Riemann Sum
Rolle's Theorem
Root Test
Second Derivative Test
Separable Equations
Separable differential equations
Separation of Variables
Simpson's Rule
Slope Fields
Solid of Revolution
Solutions to Differential Equations
Solving Inequalities using Continuity Properties
Spectral Theory
Stochastic Differential Equations
Surface Area Integral
Surface Area of Revolution
Surface Integrals
Surplus
Symmetry of Functions
Tangent Lines
Tangent Plane
Tangent Planes and Linear Approximations
Taylor Polynomials
Taylor Series
Techniques of Integration
The Fundamental Theorem of Calculus
The Limit Does Not Exist
The Mean Value Theorem
The Power Rule
The Squeeze Theorem
The Trapezoidal Rule
Theorems of Continuity
Topology
Trigonometric Substitution
Triple Integral
Triple Integral Spherical Coordinates
Triple Integrals in Cylindrical Coordinates
Types of Discontinuity
Using Slope Fields to Graph Solutions
Variational Methods
Vector Valued Function
Vectors In Space
Vectors in Calculus
Velocity as Average Rate of Change
Vertical Asymptote
Volume Integrals
Volume by disks
Volume by shells
Washer Method

Decision Maths
Discrete Mathematics
Geometry
Logic and Functions
Mechanics Maths
Probability and Statistics
Pure Maths
Statistics
Theoretical and Mathematical Physics

Contents

Applied Mathematics
Calculus

Absolute Maxima and Minima
Absolute and Conditional Convergence
Accumulation Function
Accumulation Problems
Algebraic Functions
Alternating Series
Antiderivatives
Application of Derivatives
Application of Higher Order Derivatives
Application of Integrals in Biology and Social Sciences
Applications of Continuity
Applications of Double Integrals
Approximating Areas
Arc Length of a Curve
Area Between Two Curves
Arithmetic Series
Arithmetic of Complex Numbers
Average Rate of Change of Populations
Average Value Function
Average Value of a Function
Bifurcation Theory
Boundary Value Problems
Calculus Linear Approximation
Calculus Of Parametric Curves
Candidate Test
Change of Variables in Multiple Integrals
Combining Different Rules
Combining Differentiation Rules
Combining Functions
Complex Analysis
Concavity of a Function
Continuity
Continuity Equations
Continuity Over an Interval
Continuity and Indeterminate Forms
Convergence Tests
Cost and Revenue
Critical points
Curl and Divergence
Curve Sketching Techniques
Density and Center of Mass
Derivative Functions
Derivative Properties
Derivative as a Limit
Derivative of Exponential Function
Derivative of Inverse Function
Derivative of Logarithmic Functions
Derivative of Trigonometric Functions
Derivative of Vector Valued Function
Derivatives
Derivatives and Continuity
Derivatives and the Shape of a Graph
Derivatives of Exponential Functions
Derivatives of Inverse Trigonometric Functions
Derivatives of Logarithmic Functions
Derivatives of Polar Functions
Derivatives of Sec, Csc and Cot
Derivatives of Sin, Cos and Tan
Determining Volumes by Slicing
Difference Quotient
Differentiability
Differentials
Differentiation of Functions of Several Variables
Direction Fields
Disk Method
Divergence Test
Double Integral
Double Integrals Over General Regions
Double Integrals Over Rectangular Regions
Double Integrals in Polar Coordinates
Dynamical Systems
Eliminating The Parameter
Euler's Method
Evaluating a Definite Integral
Evaluation Theorem
Exact equations
Exponential Functions
Exponential Model
Extrema
Finding Limits
Finding Limits of Specific Functions
First Derivative Test
Function Transformations
Fundamental Theorem of Line Integrals
Galois Theory
General Solution of Differential Equation
Geometric Series
Gradient Descent
Graphing and Optimization
Green's Function
Green's Theorem
Growth Rate of Functions
Harmonic Functions
Higher Order Partial Derivatives
Higher-Order Derivatives
Hydrostatic Pressure
Hyperbolic Functions
Implicit Differentiation Tangent Line
Implicit Relations
Improper Integrals
Increasing and Decreasing Functions
Indefinite Integral
Indeterminate Forms
Indeterminate Forms of Limits
Initial Value Problem Differential Equations
Integral Equations
Integral Test
Integrals in Economics
Integrals of Exponential Functions
Integrals of Motion
Integrating Even and Odd Functions
Integration Formula
Integration Tables
Integration Techniques
Integration Using Long Division
Integration Using Tables
Integration fundamentals
Integration of Logarithmic Functions
Integration of Vector Valued Functions
Integration using Inverse Trigonometric Functions
Intermediate Value Theorem
Inverse Trigonometric Functions
Jump Discontinuity
Lagrange Error Bound
Lagrange Multiplier
Laplace Transform
Lebesgue Integration
Limit Applications
Limit Laws
Limit of Vector Valued Function
Limit of a Sequence
Limits
Limits and Continuity
Limits at Infinity
Limits at Infinity and Asymptotes
Limits of a Function
Line Integrals
Linear Approximations and Differentials
Linear Differential Equation
Linear Functions
Logarithmic Differentiation
Logarithmic Functions
Logistic Differential Equation
Logistic Model
Maclaurin Series
Manipulating Functions
Matrix Theory
Maxima and Minima
Maxima and Minima Problems
Mean Value Theorem for Integrals
Measure Theory
Michaelis Menten Equation
Models for Population Growth
Motion Along A Line
Motion In Space
Multiple Integrals
Multivariable Calculus
Natural Logarithmic Function
Net Change Theorem
Newton's Method
Non Differentiable Functions
Nonhomogeneous Differential Equation
Nonlinear Differential Equations
Numerical Linear Algebra
One-Sided Limits
Optimization Problems
Optimization Problems in Economics
Optimization Theory
Ordinary Differential Equations
P Series
PDE Solutions
Parametric Surface Area
Parametric derivatives
Partial Derivatives
Particle Model Motion
Particular Solutions to Differential Equations
Piecewise Defined Function
Polar Coordinates
Polar Coordinates Functions
Polar Curves
Population Change
Power Series
Probability Theory
Properties of Definite Integrals
Radius of Convergence
Ratio Test
Real Analysis
Related Rates
Removable Discontinuity
Revenue as Average Rate of Change
Riemann Integral
Riemann Sum
Rolle's Theorem
Root Test
Second Derivative Test
Separable Equations
Separable differential equations
Separation of Variables
Simpson's Rule
Slope Fields
Solid of Revolution
Solutions to Differential Equations
Solving Inequalities using Continuity Properties
Spectral Theory
Stochastic Differential Equations
Surface Area Integral
Surface Area of Revolution
Surface Integrals
Surplus
Symmetry of Functions
Tangent Lines
Tangent Plane
Tangent Planes and Linear Approximations
Taylor Polynomials
Taylor Series
Techniques of Integration
The Fundamental Theorem of Calculus
The Limit Does Not Exist
The Mean Value Theorem
The Power Rule
The Squeeze Theorem
The Trapezoidal Rule
Theorems of Continuity
Topology
Trigonometric Substitution
Triple Integral
Triple Integral Spherical Coordinates
Triple Integrals in Cylindrical Coordinates
Types of Discontinuity
Using Slope Fields to Graph Solutions
Variational Methods
Vector Valued Function
Vectors In Space
Vectors in Calculus
Velocity as Average Rate of Change
Vertical Asymptote
Volume Integrals
Volume by disks
Volume by shells
Washer Method

Decision Maths
Discrete Mathematics
Geometry
Logic and Functions
Mechanics Maths
Probability and Statistics
Pure Maths
Statistics
Theoretical and Mathematical Physics

Contents

Fact Checked Content
Last Updated: 22.07.2022
11 min reading time

Content creation process designed by Lily Hulatt
Content sources cross-checked by Gabriel Freitas
Content quality checked by Gabriel Freitas

Jump to a key chapter

Forming Riemann sums area parabola rectangles StudySmarter Approximation of the area below a curve using rectangles - StudySmarter Originals

The first step is to divide the interval of integration into as many subintervals as we want. Remember that the more subintervals we use, the better approximation we get. Right-endpoint and left-endpoint approximations use the very points we obtain when making a partition to find the height of each rectangle. However, we are not limited to these points!

Sigma notation formulas

Before proceeding, we will introduce some formulas in Sigma Notation that will make our lives easier when working with sums.

The following expressions give the sum of consecutive integers, square of integers, and cube of integers, respectively:

∑i=1ni=n(n+1)2

∑i=1ni2=n(n+1)(2n+1)6

∑i=1ni3=n(n+1)22

The above formulas, along with basic properties of summations, are very useful when approximating areas using sigma notation. Let's see an example using them!

Evaluate the following sum: S=∑i=110(i2+2i-1)

Let's use the properties of summation and the above formulas to evaluate the given sum!

Use the properties of summation to rewrite the sum.

S=∑i=110i2-2∑ii=110-∑1i=110

Use ∑i=1ni2=n(n+1)(2n+1)6 with n=10 to evaluate the first sum.

S=10(10+1)(2(10)+1)6-2∑i=110i-∑1i=110

Simplify.

S=385-2∑i=110i-∑i=1101

Use ∑i=110i=n(n+1)2with n=10 to evaluate the second sum.

S=385-10(10+1)2-∑i=1101

Simplify.

S=385-55-∑i=1101

The last sum is just adding 1 together 10 times, which is the same as 10 times 1.

S=385-55-10

Simplify.

S=320

Forming Riemann Sums

One way to approximate the area below a curve is to divide the area into rectangles. We get this done in the following way:

Begin by dividing the interval of the area into subintervals. The more, the better!
Associate a rectangle to each subinterval.
The width of each rectangle is equal to the length of its corresponding subinterval.
The height of each rectangle can be found by evaluating the function at some point within the subinterval.
- The left-endpoint approximation uses the leftmost value of the subinterval.
- The right-endpoint approximation uses the rightmost value of the subinterval.

But what limits our choice to just the leftmost value or the rightmost value? Nothing! As long as we take any point within the interval, we are good to go!

Of course our approximations will differ, but we should focus on increasing the number of intervals rather than worrying about which point to use for the height of the rectangles.

Riemann sum definition

After dividing the area below a curve into rectangles, we add them up, and we get an approximation for the area. This is known as a Riemann Sum, named after the mathematician Bernhard Riemann, who worked on the idea in the 19th century.

We need to set up some background before defining a Riemann Sum. Let f(x)be a function defined on a closed interval a≤x≤band let P={xi}for i=0, 1, 2, ..., nbe a regular partition of a≤x≤b. We have a set of n subintervals of the form xi-1≤x≤xi, all with a length of Δx=b-an.

Forming Riemann sums regualar partition StudySmarter A regular partition of the interval - StudySmarter Originals

Up to this point, we have the same set-up for the right-endpoint and left-endpoint approximations using equally-spaced subintervals. The difference comes from which point we use to find the height of the rectangles.

A Riemann Sum for the function f(x)for the partition P is defined as:

∑i=1nf(xi*)Δx

Where xi*is any value inside the interval xi-1≤x≤xi, with \(x_i \in P\).

The right-endpoint and left-endpoint approximations are particular cases of Riemann Sums.

Riemann sums and midpoint approximation

As we can use any value inside each subinterval when finding Riemann sums, why not use the midpoint? This is known as the midpoint approximation.

The midpoint approximation Mn of the area below a curve is a particular case of a Riemann sum obtained by choosing the midpoint of each subinterval. That is:

Mn=∑i=1nfxi-1+xi2Δx

As usual, this is better understood with an example. Let's take a look at a midpoint approximation!

Use a midpoint approximation to approximate the area below f(x)=x2+1in the interval 0≤x≤2by dividing the interval into 10subintervals of the same size.

Find the length of each subinterval.

Δx=b-an=2-010=0.2

Since the length of each subinterval is 0.2, our partition will be made of the points 0, 0.2, 0.4, ..., 1.6, 1.8, and 2. This can be summarized as follows:

xi=0.2i

We are using the midpoint approximation, so the midpoint between each two values of the partition will be used to find the height of the rectangles.

Use the formula for the midpoint approximation with n=10 and Δx=0.2.

M10=∑i=110fxi-1+xi2(0.2)

Rather than writing xi-1+xi2we can use the expression for xiand simplify.

xi-1+xi2=0.2(i-1)+0.2i2=0.2i-0.1

We can substitute back this expression into our formula and use the properties of summation to find our approximation.

M10=∑i=110f(0.2i-0.1)(0.2)

Factor out 0.2 from the summation.

M10=0.2∑i=110f(0.2i-0.1)

Evaluate the function at 0.2i-0.1

M10=0.2∑i=110(0.2i-0.1)2+1

Expand the binomial and simplify.

M10=0.2∑i=1100.04i2-0.04i+1.01

Use the properties of summation to rewrite the sum.

M10=0.008∑i=110i2-0.008∑i=110i+0.202∑i=1101

Use ∑i=1ni2=n(n+1)(2n+1)6 to evaluate the first sum.

M10=0.00810(11)(21)6-0.008∑i=110i+0.202∑i=1101

Use ∑i=1ni=n(n+1)2to evaluate the second sum.

M10=0.00810(11)(21)6-0.00810(11)2+0.202∑i=1101

Evaluate the last sum.

M10=0.00810(11)(21)6-0.00810(11)2+2.02

Evaluate using a calculator.

M10=4.66

Remember that we get better approximations as we divide the interval into more subintervals!

Until now, we have been approximating areas without knowing which approximation is better. Is there a way of telling which Riemann sum is better? Sadly, if we do not know the real value of the area below the curve, there is no telling which approximation is better. Besides, it would be pointless to do an approximation while knowing the actual value!

However, there is a way of squeezing the value of the area between two values.

Upper Riemann sum

We are allowed to take any value inside each subinterval to form our Riemann sum. What if we take the maximum value of f

in each subinterval? The Riemann sum obtained through this process is called an upper Riemann sum.

A Riemann sum for the function f(x) for the partition P={xi} is called an upper Riemann sum if the values xi*are taken as the maximum value from each subinterval.

In this case, we can guarantee that our approximation will be greater than or equal to the actual value of the area since we are taking bigger pieces of the area! If we let A be the area below the curve and SUbe an upper Riemann sum, then we can write the following inequality:

SU≥A

Let's now find an upper Riemann sum of the function of our previous example, f(x)=x2+1, in the same interval 0≤x≤2 by dividing the interval into 10 subintervals as well. We will begin by taking a look at its graph.

Forming Riemann sums parabola StudySmarter Graph of the parabola f(x) divided with a left-endpoint approximation - StudySmarter Originals

We can notice how this function is an increasing function, hence, the greatest value from each subinterval is the rightmost value. Therefore, doing a right-endpoint approximation will give us an upper Riemann sum.

Use the formula for the right-endpoint approximation.

R10=∑i=110f(xi)Δx

Substitute xi=iΔx and Δx=2-010=0.2into the formula.

R10=∑i=110f(0.2i)0.2

Factor out 0.2 and evaluate the function.

R10=0.2∑i=110((0.2i)2+1)

Use the properties of summation to rewrite the sum.

R10=0.008∑i=110i2+0.2∑i=1101

Use ∑i=1ni2=n(n+1)(2n+1)6to evaluate the first sum.

R10=0.00810(11)(21)6+0.2∑i=1101

Evaluate the last sum.

R10=0.00810(11)(21)6+0.2(10)

Evaluate using a calculator.

R10=5.08

We can see that the value of our approximation is greater than the midpoint approximation. We also know that this value is greater than the actual area below the curve!

Lower Riemann sum

And what if we take the minimum value of f in

each subinterval? We get a lower Riemann sum!

A Riemann sum for the function f(x) for the partition P={xi} is called a lower Riemann sum if the values xi*are taken as the minimum value from each subinterval.

This time, we can guarantee that our approximation will be less than or equal to the actual value of the area since we are taking smaller pieces of the area! If we let A be the area below the curve and SLbe a lower Riemann sum, then we can write the following inequality:

SL≤A

It's time to find a lower Riemann sum of our previous examples. In this case, the least value from each subinterval is the leftmost value, so a left-endpoint approximation will give us a lower Riemann sum.

Use the formula for the left-endpoint approximation.

L10=∑i=09f(xi)Δx

Substitute xi=iΔxand Δx=2-010=0.2into the formula.

L10=∑i=09f(0.2i)0.2

Factor out 0.2 and evaluate the function.

L10=0.2∑i=09((0.2i)2+1)

Use the properties of summation to rewrite the sum.

L10=0.008∑i=09i2+0.2∑i=091

Note that the summations start at 0. The last sum still consists of 10 terms, so it can be evaluated as adding 1 ten times together.

Evaluate the last sum.

L10=0.008∑i=09i2+0.2(10)

For the first sum note that the term containing i=0 does not contribute to the sum because 02=0, so this sum can be evaluated as if it was from i=1 to 9.

Use ∑i=1ni2=n(n+1)(2n+1)6 to evaluate the sum.

L10=0.0089(10)(19)6+0.2(10)

Evaluate using a calculator.

L10=4.28

We got an approximation of 4.28. This means that the actual value of the area below the curve lies between 4.28 and 5.08!

Riemann sums and area below a curve

We have seen how lower and upper Riemann sums give us a bound on the area below a curve. This also depends on how many subintervals we divide the area below the curve. What if we use an infinite amount of subintervals?

Let f(x)be a continuous function on an interval a≤x≤b. Then, the area below the curve is obtained as the limit as n goes to infinity of a Riemann sum:

A=limn→∞ ∑i=1nf(xi*)Δx

Where xi*is any value inside the interval xi-1≤x≤xiand Δx=b-an.

Let's think of the above expression: as we increase n, we get thinner rectangles, which will then perfectly fit below the curve. By adding up all these rectangles we get the area below the curve! Let's take a look at an illustration using 70 rectangles.

Forming Riemann sums parabola thin rectangles StudySmarter Approximation of the area below the curve using 70 rectangles - StudySmarter Originals

This seems to fit very well the area below the curve, right? Now imagine what would happen if we used even more rectangles!

Summary

Forming Riemann Sums - Key takeaways

A Riemann sum consists of dividing the area below a curve into rectangles and adding them up.
- Riemann sums are closely related to the left-endpoint and right-endpoint approximations. Both are particular cases of a Riemann sum.
A lower Riemann sum is a Riemann sum obtained by using the least value of each subinterval to calculate the height of each rectangle.
- The value of a lower Riemann sum is always less than or equal to the area below the curve.
An upper Riemann sum is a Riemann sum obtained by using the greatest value of each subinterval to calculate the height of each rectangle.
- The value of an upper Riemann sum is always greater than or equal to the area below the curve.
The area below a curve is bounded between a lower Riemann sum and an upper Riemann sum.
If we take the limit of a Riemann sum as the number of subintervals tends to infinity we get the area below the curve.

Flashcards in Riemann Sum

6 Start learning

All Riemann sums are either a left-endpoint approximation or a right-endpoint approximation.

False.

The right-endpoint approximation is a particular case of a Riemann sum.

True.

An upper Riemann sum is always greater than or equal to the exact area below the curve.

True.

A lower Riemann sum is always less than or equal to the exact area below the curve.

True.

Suppose that, in some interval, a lower Riemann sum of a function is 24.92 and an upper Riemann sum is 25.08. What can we conclude from this information?

That the area below the curve is between 24.92 and 25.08.

A left-endpoint approximation always gives us a lower Riemann sum.

False.

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Frequently Asked Questions about Riemann Sum

What is a Riemann sum?

A Riemann sum consists of dividing the area below a curve into rectangles and adding them up.

How do you find the Riemann sum of a function?

Begin by dividing the interval of the area into subintervals. The more, the better!
Associate a rectangle to each subinterval.
The width of each rectangle is equal to the length of its corresponding subinterval.
The height of each rectangle can be found by evaluating the function at some point within the subinterval.

Usually, the interval is divided regularly. This means that every subinterval will have the same length.

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All Riemann sums are either a left-endpoint approximation or a right-endpoint approximation.

A. True. B. False.

The right-endpoint approximation is a particular case of a Riemann sum.

A. True. B. False.

An upper Riemann sum is always greater than or equal to the exact area below the curve.

A. True. B. False.

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