8. Definite Integrals

Riemann Sums

8. Definite Integrals

Riemann Sums: Videos & Practice Problems

Topic summary

Understanding sigma notation is essential for evaluating finite sums efficiently. It allows for compact representation of summations, such as $S^{n} = \sum k_{1} f^{(x k_{- 1})}$ . Riemann sums, which estimate the area under curves, can be calculated using left, right, or midpoint evaluations, enhancing the understanding of functions and their applications in calculus.

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Sigma Notation

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Sigma Notation Video Summary

When tasked with summing a series of numbers, such as 1 + 2 + 3 + 4, it can be manageable. However, extending this to larger sequences, like adding numbers up to 100, becomes cumbersome. To simplify this process, we can utilize sigma notation, a compact way to represent the sum of a sequence of terms.

Sigma notation is represented by the Greek letter Sigma (Σ) and allows us to express the sum of a series of values efficiently. The notation includes an index of summation, which indicates the starting point of the summation, and an upper limit that shows where the summation ends. For example, in the expression:

$$\sum_{k=1}^{n} f(k)$$

the index starts at 1 and goes up to n, where f(k) is the function applied to each integer value of k. This means you will evaluate the function for each integer from the starting index to the ending index and sum the results.

To illustrate, consider the finite sum where we evaluate:

$$\sum_{k=1}^{3} k^2$$

In this case, we substitute k with the integers from 1 to 3:

$$1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14$$

Thus, the result of this summation is 14. The process is straightforward: plug in the integer values into the function and sum the results.

It's important to note that the index of summation can vary; it doesn't have to be k. It could be represented by any letter, such as i, j, or h. For example, if we have:

$$\sum_{i=0}^{4} (i + 3)$$

we would evaluate this from i = 0 to i = 4:

$$0 + 3 + 1 + 3 + 2 + 3 + 3 + 3 + 4 + 3 = 25$$

Thus, the result of this summation is 25. Understanding sigma notation allows for efficient computation of sums, especially when dealing with larger sequences, and is a foundational concept in calculus, particularly in approximating areas under curves using rectangles.

Problem

Evaluate the following summation:
$\sum_{i=1}^4i^2-3i+8$

Problem

Evaluate the following summation:
$\sum_{k=1}^4\left(\frac{k}{_{}2}\right)^2$

$\frac{15}{2}$

$15$

$\frac{17}{2}$

$\frac{17}{4}$

Problem

Evaluate the following summation:
$\sum_{i=0}^2\frac{2i}{3}$

$6$

$\frac23$

$2$

$\frac43$

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Sigma Notation Example 1

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Sigma Notation Example 1 Video Summary

In this discussion, we explore the process of converting expanded sums into sigma notation, a concise way to represent summations. The key to this transformation lies in recognizing patterns within the given sums. Let's break down the approach using two examples.

Starting with the first example, we observe a consistent factor of three in the sum, indicating that this number will be part of our sigma notation. Additionally, each term in the sum involves a variable that is squared, specifically $ k^2 $. The values of $ k $ range from 1 to 5, suggesting that our summation will start at $ k = 1 $ and end at $ k = 5 $. Therefore, the sigma notation for this sum can be expressed as:

$$ \sum_{k=1}^{5} 3k^2 $$

This notation effectively captures the repetitive nature of the original expanded sum.

Moving on to the second example, we encounter a series of fractions. The first term is $ \frac{3}{4} $, followed by $ \frac{3}{2} $. To simplify the representation, we can convert $ \frac{3}{2} $ into a fraction with a common denominator of 4, resulting in $ \frac{6}{4} $. The subsequent terms, $ \frac{9}{4} $ and $ \frac{12}{4} $, follow a similar pattern. Here, we notice that the numerator is consistently a multiple of 3, while the denominator remains 4.

To express this in sigma notation, we start with the constant factor $ \frac{3}{4} $ outside the summation. The variable $ k $ begins at 1 and increases, with the numerator being $ 3k $. The upper limit of the summation is 4, as there are four terms in total. Thus, the sigma notation for this example is:

$$ \frac{3}{4} \sum_{k=1}^{4} k $$

In summary, converting expanded sums into sigma notation involves identifying constant factors, recognizing patterns in the variable terms, and determining the appropriate limits for the summation. This method not only simplifies the representation of sums but also enhances our understanding of the underlying mathematical relationships.

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Algebra Rules for Finite Sums

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Algebra Rules for Finite Sums Video Summary

In the study of summation, sigma notation serves as a powerful tool for representing finite sums. Understanding the rules that govern these summations is crucial for simplifying and solving problems efficiently. Key among these rules are the sum and difference rule, the constant multiple rule, and the treatment of constant summations.

The sum and difference rule allows us to break down a summation involving addition or subtraction into separate sums. For instance, if we have a summation of the form $\sum (a_k \pm b_k)$, we can express it as $\sum a_k \pm \sum b_k$. This approach mirrors the techniques used in calculus for derivatives and integrals, making it easier to manage complex expressions. For example, evaluating the summation from $k=1$ to $3$ of $k^2 - k$ can be simplified to $\sum k^2 - \sum k$, yielding results of $14$ and $6$ respectively, leading to a final answer of $8$.

The constant multiple rule states that if a constant $c$ multiplies a summation, it can be factored out. For example, $\sum_{k=1}^{n} c \cdot a_k = c \cdot \sum_{k=1}^{n} a_k$. This simplifies calculations significantly. If we consider the summation $\sum_{k=1}^{3} 2k^2$, we can factor out the $2$ to get $2 \cdot \sum_{k=1}^{3} k^2$, which simplifies to $2 \cdot 14 = 28$.

When summing a constant value, the result can be calculated by multiplying the constant by the number of terms. For instance, $\sum_{k=1}^{n} c = c \cdot n$. If we evaluate $\sum_{k=1}^{18} 3$, it simplifies to $3 \cdot 18 = 54$.

In practice, many problems will require the application of multiple rules simultaneously. For example, to evaluate the sum $\sum_{k=1}^{3} (6k^2 - 9)$, we can apply the sum and difference rule to separate it into $6 \sum_{k=1}^{3} k^2 - 9 \sum_{k=1}^{3} 1$. This leads to $6 \cdot 14 - 9 \cdot 3$, resulting in $57$ as the final answer.

By mastering these rules, students can approach summation problems with confidence and efficiency, leveraging the structure of sigma notation to simplify their calculations.

Problem

Evaluate the following summation:
$\sum_{t=2}^65t+\frac{4}{t}$

$\frac{471}{5}$

$\frac{529}{5}$

$\frac{26}{3}$

$\frac{22}{3}$

Problem

Evaluate the following summation:
$\sum_{i=0}^42i+1$

Problem

Evaluate the following summation:
$\sum_{i=1}^24i^3-3i^2+2i-1$

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Introduction to Riemann Sums

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Introduction to Riemann Sums Video Summary

In the study of calculus, Riemann sums provide a method for estimating the area under a curve by dividing it into smaller sections, typically rectangles. This approach combines the concepts of finite sums and area estimation, allowing for a more systematic calculation of integrals. The fundamental idea is to approximate the area under a function by summing the areas of these rectangles, which can be expressed using summation notation.

To begin, we define the width of each rectangle, known as delta x (Δx), which is calculated using the formula:

$$\Delta x = \frac{b - a}{n}$$

where a and b are the endpoints of the interval, and n is the number of rectangles. For example, if we want to estimate the area under a curve from 0 to 2 using four rectangles, we would find:

$$\Delta x = \frac{2 - 0}{4} = 0.5$$

Next, we can express the Riemann sum using sigma notation. For left endpoints, the Riemann sum can be written as:

$$R_n = \sum_{k=1}^{n} f(x_{k-1}) \Delta x$$

In this notation, f(x_k-1) represents the function evaluated at the left endpoint of each subinterval, and the summation runs from k = 1 to n.

For our example with four rectangles, the Riemann sum would be:

$$R_4 = \Delta x \sum_{k=1}^{4} f(x_{k-1})$$

Substituting the values, we evaluate the function at the left endpoints: f(0), f(0.5), f(1), and f(1.5). The total area can then be approximated by multiplying the sum of these function values by delta x.

In general, Riemann sums can also be calculated using right endpoints or midpoints, leading to a more versatile approach. The notation for a general Riemann sum is:

$$R_n = \sum_{k=1}^{n} f(x_k^*) \Delta x$$

Here, x_k^* indicates that the evaluation point can be any point within the subinterval, allowing for flexibility in the estimation process.

By mastering Riemann sums, students can effectively approximate the area under curves, laying the groundwork for understanding definite integrals and the fundamental theorem of calculus.

Problem

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,5] with 5 subintervals.

$5$

$\frac{15}{2}$

$6.25$

Problem

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,6] with 6 subintervals.

9.62

6.62

7.16

8.15

Problem

Write the Riemann sum that would approximate the area of the following graph over the interval [0,3] using 3 subintervals.

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Left, Right, & Midpoint Riemann Sums

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Left, Right, & Midpoint Riemann Sums Video Summary

Riemann sums provide a method for approximating the area under a curve by using rectangles. This technique can be applied using left endpoints, right endpoints, or midpoints of the rectangles, each yielding different approximations of the area. Understanding the notation and setup for each type of Riemann sum is crucial for effectively applying this concept.

To begin, the left endpoint Riemann sum is calculated by evaluating the function at the leftmost point of each rectangle. The formula for the left endpoint Riemann sum, denoted as $ L_n $, is given by:

L_n = \sum_{k=1}^{n} f(x_{k-1}) \Delta x

Here, $ \Delta x $ represents the width of each rectangle, calculated as:

\Delta x = \frac{b - a}{n}

where $ a $ and $ b $ are the bounds of the interval, and $ n $ is the number of rectangles. For example, if the interval is from 0 to 2 and $ n = 4 $, then $ \Delta x = \frac{2 - 0}{4} = 0.5 $.

In contrast, the right endpoint Riemann sum uses the rightmost point of each rectangle. The formula for the right endpoint Riemann sum, denoted as $ R_n $, is:

R_n = \sum_{k=1}^{n} f(x_k) \Delta x

In this case, the function is evaluated at $ x_k $, which is the right endpoint of each rectangle. The calculation of $ \Delta x $ remains the same as in the left endpoint case.

Midpoint Riemann sums provide another approximation method by evaluating the function at the midpoint of each rectangle. The formula for the midpoint Riemann sum, denoted as $ M_n $, is expressed as:

M_n = \sum_{k=1}^{n} f\left(\frac{x_{k-1} + x_k}{2}\right) \Delta x

This formula averages the left and right endpoints to find the midpoint, which is then used to evaluate the function. The width $ \Delta x $ remains consistent across all types of Riemann sums.

In summary, Riemann sums can be calculated using left endpoints, right endpoints, or midpoints, each providing a different approximation of the area under a curve. The key is to understand how to set up the summation and apply the appropriate function evaluations based on the chosen method. Mastery of this concept is essential for further studies in calculus and integral approximations.

Problem

Approximate the area under the curve $f\left(x\right)=\sqrt{x+3}$ over the interval $\left\lbrack1,5\right\rbrack$ using the Right Riemann sum with 8 subintervals.

9.75

9.96

9.54

9.72

Problem

Approximate the area under the curve $f\left(x\right)=\frac{2}{3x+5}$ over the interval $\left\lbrack4,10\right\rbrack$ using the Midpoint Riemann sum with $n=3$ subintervals.

0.55

0.48

0.43

0.61

example

Left, Right, & Midpoint Riemann Sums Example 1

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10m

Left, Right, & Midpoint Riemann Sums Example 1 Video Summary

To solve the problem of estimating the area under the curve of the function $ f(x) = x^3 $ over the interval from 0 to 4 using Riemann sums, we will utilize left, right, and midpoint approximations with four subintervals.

First, we calculate the width of each subinterval, denoted as $ \Delta x $. This is determined by the formula:

\[\Delta x = \frac{b - a}{n}\]

where $ b = 4 $, $ a = 0 $, and $ n = 4 $. Thus, we have:

\[\Delta x = \frac{4 - 0}{4} = 1\]

Next, we will compute the left endpoint Riemann sum, denoted as $ L_n $. The formula for the left endpoint sum is:

\[L_n = \sum_{k=1}^{n} f(x_{k-1}) \Delta x\]

Substituting our values, we find:

\[L_n = \sum_{k=1}^{4} f(k-1) \cdot 1 = f(0) + f(1) + f(2) + f(3)\]

Calculating each term:

$ f(0) = 0^3 = 0 $
$ f(1) = 1^3 = 1 $
$ f(2) = 2^3 = 8 $
$ f(3) = 3^3 = 27 $

Thus, the left endpoint sum is:

\[L_n = 0 + 1 + 8 + 27 = 36\]

Now, we compute the right endpoint Riemann sum, denoted as $ R_n $. The formula for the right endpoint sum is:

\[R_n = \sum_{k=1}^{n} f(x_k) \Delta x\]

Substituting our values, we find:

\[R_n = \sum_{k=1}^{4} f(k) \cdot 1 = f(1) + f(2) + f(3) + f(4)\]

Calculating each term:

$ f(1) = 1^3 = 1 $
$ f(2) = 2^3 = 8 $
$ f(3) = 3^3 = 27 $
$ f(4) = 4^3 = 64 $

Thus, the right endpoint sum is:

\[R_n = 1 + 8 + 27 + 64 = 100\]

Finally, we compute the midpoint Riemann sum, denoted as $ M_n $. The formula for the midpoint sum is:

\[M_n = \sum_{k=1}^{n} f\left(\frac{x_{k-1} + x_k}{2}\right) \Delta x\]

Substituting our values, we find:

\[M_n = \sum_{k=1}^{4} f\left(\frac{(k-1) + k}{2}\right) \cdot 1\]

Calculating the midpoints:

For $ k=1 $: $ f(0.5) = (0.5)^3 = 0.125 $
For $ k=2 $: $ f(1.5) = (1.5)^3 = 3.375 $
For $ k=3 $: $ f(2.5) = (2.5)^3 = 15.625 $
For $ k=4 $: $ f(3.5) = (3.5)^3 = 42.875 $

Thus, the midpoint sum is:

\[M_n = 0.125 + 3.375 + 15.625 + 42.875 = 62\]

In summary, the results of our Riemann sums are:

Left endpoint approximation: $ L_n = 36 $
Right endpoint approximation: $ R_n = 100 $
Midpoint approximation: $ M_n = 62 $

To determine which sums are overestimates and which are underestimates, we analyze the results. The left endpoint sum of 36 is an underestimate, as it provides a lower area than the actual area under the curve. Conversely, the right endpoint sum of 100 is an overestimate, as it exceeds the actual area. The midpoint sum of 62 offers a more accurate approximation, falling between the left and right estimates.

In conclusion, the left endpoint provides an underestimate, the right endpoint provides an overestimate, and the midpoint gives a more accurate estimate of the area under the curve of the function $ f(x) = x^3 $ over the interval from 0 to 4.

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Here’s what students ask on this topic:

What is sigma notation and how is it used in calculus?

Sigma notation is a mathematical symbol used to represent the sum of a sequence of numbers. It is denoted by the Greek letter Σ and allows for a compact representation of summations. In calculus, sigma notation is essential for evaluating finite sums efficiently, such as $\sum k_{1} k_{2}$ . It simplifies the process of adding a series of terms by providing a clear structure for the summation process. Understanding sigma notation is crucial for solving problems related to integration and area under curves, as it forms the basis for Riemann sums, which estimate areas using rectangles. Mastery of sigma notation enhances comprehension of integral calculus concepts.

How do you evaluate a finite sum using sigma notation?

To evaluate a finite sum using sigma notation, you start by identifying the sequence of numbers to be summed and the function that represents them. The notation $\sum k_{1} k_{2}$ indicates the sum of terms from the index of summation, k₁, to k₂. You plug in integer values from the start to the end of the index into the function, then add the results. For example, if the function is k² and the index is from 1 to 3, you calculate 1² + 2² + 3², resulting in 14. This process is straightforward once you understand the notation and the function involved.

What are Riemann sums and how are they used in calculus?

Riemann sums are a method used in calculus to approximate the area under a curve by dividing it into rectangles. They are calculated using sigma notation, which provides a compact way to sum the areas of these rectangles. Riemann sums can be evaluated using left, right, or midpoint methods, depending on which part of the rectangle is used to determine the height. The formula involves summing the product of the function value at a specific point and the width of the subinterval. This technique is fundamental in understanding integration, as it provides a practical approach to estimating areas and understanding the concept of definite integrals.

What are the rules for summation in sigma notation?

In sigma notation, several rules simplify the evaluation of summations. The sum and difference rule allows you to separate terms within a summation, similar to how you handle derivatives and integrals. The constant multiple rule lets you factor out constants from the summation, making calculations more straightforward. Additionally, when summing a constant value, you multiply the constant by the number of terms in the index. These rules streamline the process of evaluating finite sums and are analogous to rules used in differentiation and integration, providing a consistent framework for handling mathematical operations in calculus.

How do you set up a Riemann sum using left, right, and midpoint evaluations?

To set up a Riemann sum, you first decide whether to use left, right, or midpoint evaluations. For left endpoints, the function value is calculated at the start of each subinterval. For right endpoints, it is calculated at the end. Midpoint evaluations use the average of the left and right endpoints. The general formula involves summing the product of the function value at the chosen point and the width of the subinterval, $\sum k_{1} k_{n}$ f(x_k)Δx. This setup allows for a systematic approach to approximating the area under a curve, essential for understanding integration.

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Riemann Sums: Videos & Practice Problems

Sigma Notation

Sigma Notation Video Summary

Evaluate the following summation: ∑i=14i2−3i+8\sum_{i=1}^4i^2-3i+8∑i=14i2−3i+8

Evaluate the following summation: ∑k=14(k2)2\sum_{k=1}^4\left(\frac{k}{_{}2}\right)^2∑k=14(2k)2

Evaluate the following summation: ∑i=022i3\sum_{i=0}^2\frac{2i}{3}∑i=0232i

Sigma Notation Example 1

Sigma Notation Example 1 Video Summary

Algebra Rules for Finite Sums

Algebra Rules for Finite Sums Video Summary

Evaluate the following summation: ∑t=265t+4t\sum_{t=2}^65t+\frac{4}{t}∑t=265t+t4

Evaluate the following summation: ∑i=042i+1\sum_{i=0}^42i+1∑i=042i+1

Evaluate the following summation: ∑i=124i3−3i2+2i−1\sum_{i=1}^24i^3-3i^2+2i-1∑i=124i3−3i2+2i−1

Introduction to Riemann Sums

Introduction to Riemann Sums Video Summary

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,5] with 5 subintervals.

For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,6] with 6 subintervals.

Write the Riemann sum that would approximate the area of the following graph over the interval [0,3] using 3 subintervals.

Left, Right, & Midpoint Riemann Sums

Left, Right, & Midpoint Riemann Sums Video Summary

Approximate the area under the curve f(x)=x+3f\left(x\right)=\sqrt{x+3}f(x)=x+3 over the interval [1,5]\left\lbrack1,5\right\rbrack[1,5] using the Right Riemann sum with 8 subintervals.

Approximate the area under the curve f(x)=23x+5f\left(x\right)=\frac{2}{3x+5}f(x)=3x+52 over the interval [4,10]\left\lbrack4,10\right\rbrack[4,10] using the Midpoint Riemann sum with n=3n=3n=3 subintervals.

Left, Right, & Midpoint Riemann Sums Example 1

Left, Right, & Midpoint Riemann Sums Example 1 Video Summary

Do you want more practice?

Here’s what students ask on this topic:

What is sigma notation and how is it used in calculus?

How do you evaluate a finite sum using sigma notation?

What are Riemann sums and how are they used in calculus?

What are the rules for summation in sigma notation?

How do you set up a Riemann sum using left, right, and midpoint evaluations?

Your Business Calculus tutors

Evaluate the following summation:
$\sum_{i=1}^4i^2-3i+8$

Evaluate the following summation:
$\sum_{k=1}^4\left(\frac{k}{_{}2}\right)^2$

Evaluate the following summation:
$\sum_{i=0}^2\frac{2i}{3}$

Evaluate the following summation:
$\sum_{t=2}^65t+\frac{4}{t}$

Evaluate the following summation:
$\sum_{i=0}^42i+1$

Evaluate the following summation:
$\sum_{i=1}^24i^3-3i^2+2i-1$

Approximate the area under the curve $f\left(x\right)=\sqrt{x+3}$ over the interval $\left\lbrack1,5\right\rbrack$ using the Right Riemann sum with 8 subintervals.

Approximate the area under the curve $f\left(x\right)=\frac{2}{3x+5}$ over the interval $\left\lbrack4,10\right\rbrack$ using the Midpoint Riemann sum with $n=3$ subintervals.