8. Definite Integrals

Riemann Sums

8. Definite Integrals

Riemann Sums: Videos & Practice Problems

Topic summary

Understanding sigma notation is essential for evaluating finite sums, which can simplify complex calculations. The sum and difference rule allows separation of terms, while the constant multiple rule enables pulling constants outside the summation. Riemann sums, which estimate the area under curves, can be calculated using left, right, or midpoint approximations, with the general formula $S = \sum f_{x * k} Δx$ . Mastering these concepts enhances problem-solving efficiency in calculus.

concept

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 express the sum of a sequence of terms.

Sigma notation is represented by the Greek letter Sigma (Σ) and allows us to denote the sum of a series of values efficiently. The notation consists of an index of summation, which indicates the starting point, 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 substitute k with each integer from 1 to n and sum the results.

To illustrate, consider the finite sum where we evaluate:

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

This means we will calculate:

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

Thus, the result of this summation is 14. The process involves substituting the index value into the function and summing the results until reaching the upper limit.

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 it as follows:

$$0 + 3 + 1 + 3 + 2 + 3 + 3 + 3 + 4 + 3$$

Calculating this gives:

$$3 + 4 + 5 + 6 + 7 = 25$$

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

Problem

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

$\frac{15}{2}$

$15$

$\frac{17}{2}$

$\frac{17}{4}$

Problem

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

$6$

$\frac{2}{3}$

$2$

$\frac{4}{3}$

Problem

Evaluate the following summation (make sure your calculator is in radian mode):
$\sum_{i = 1}^{4} \sin (\frac{π i}{8})$

1.00

1.38

3.01

0.62

example

Sigma Notation Example 1

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Sigma Notation Example 1 Video Summary

In mathematics, converting expanded sums into sigma notation involves recognizing patterns in the sequences of numbers. Sigma notation, represented by the Greek letter sigma (Σ), allows us to express sums concisely. To illustrate this process, we can analyze two examples.

In the first example, we observe a consistent pattern where the number three remains constant, while the variable $ k $ is squared. The sequence starts at $ k = 1 $ and continues up to $ k = 5 $. This leads us to express the sum in sigma notation as:

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

This notation succinctly captures the repetitive operation of squaring $ k $ and multiplying by three for each integer value from 1 to 5.

Moving on to the second example, we deal with fractions. The first term is $ \frac{3}{4} $, followed by $ \frac{3}{2} $, which can be rewritten with a common denominator of 4 as $ \frac{6}{4} $. The subsequent terms are $ \frac{9}{4} $ and $ \frac{12}{4} $. Here, we notice that the numerator is a multiple of three, while the denominator remains four. By letting $ k $ start at 1, we can express the terms as follows:

For $ k = 1 $: $ \frac{3 \cdot 1}{4} = \frac{3}{4} $

For $ k = 2 $: $ \frac{3 \cdot 2}{4} = \frac{6}{4} $

For $ k = 3 $: $ \frac{3 \cdot 3}{4} = \frac{9}{4} $

For $ k = 4 $: $ \frac{3 \cdot 4}{4} = \frac{12}{4} $

Thus, the sigma notation for this sum can be expressed as:

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

In summary, converting expanded sums into sigma notation requires identifying the constant factors and the variable patterns. By recognizing these elements, we can effectively condense the sums into a more manageable form, facilitating easier calculations and clearer communication of mathematical ideas.

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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 Σ(a_k ± b_k), we can express this as Σa_k ± Σb_k. This is similar to how we handle derivatives and integrals, where we can separate terms for easier computation. For example, evaluating the summation from k=1 to 3 of (k² - k) involves calculating Σk² and Σk separately, yielding results that can be combined for the final answer.

Another important rule is the constant multiple rule, which states that if a constant is multiplied by a function within a summation, it can be factored out. For example, Σ(c * a_k) can be rewritten as c * Σa_k. This simplification can significantly reduce the complexity of calculations. For instance, if we have Σ(2k²) from k=1 to 3, we can factor out the 2, leading to 2 * Σ(k²), which is easier to compute.

When dealing with the summation of a constant, such as Σc from k=1 to n, the result is simply c multiplied by n. For example, Σ3 from k=1 to 18 equals 3 * 18, which simplifies to 54. This rule streamlines the process of summing constant values over a range.

In practice, many problems will require the application of multiple rules in conjunction. For instance, when evaluating a more complex sum like Σ(6k² - 9) from k=1 to 3, one would first apply the sum and difference rule to separate the terms, then use the constant multiple rule to factor out constants, and finally compute the individual summations. This systematic approach leads to a final result of 57 for the given example.

By mastering these rules, students can efficiently tackle a wide range of summation problems, enhancing their mathematical problem-solving skills and understanding of series and sequences.

Problem

Evaluate the following summation:
$\sum_{t = 2}^{6} 5 t + \frac{4}{t}$

$471 over 5$

$529 over 5$

$26 over 3$

$22 over 3$

Problem

Evaluate the following summation:
$\sum_{i = 0}^{4} 2 i + 1$

Problem

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

Problem

Evaluate the following summation (make sure your solution is in radians):
$\sum_{k = 0}^{5} 2 \tan (\frac{π k}{3}) - π k$

-43.65

-47.12

-50.58

concept

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 x = 0 to x = 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 Δx is the width of the rectangles. This allows us to compactly represent the sum of the areas of the rectangles.

For instance, if we are estimating the area under a function using four rectangles, we would evaluate the function at the left endpoints, which might be f(0), f(0.5), f(1), and f(1.5). The total area can then be approximated by summing these values multiplied by Δx.

In general, the Riemann sum can be adapted to use right endpoints or midpoints as well. The notation for a more 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 method.

Ultimately, Riemann sums serve as a foundational concept in integral calculus, bridging the gap between discrete sums and continuous area calculations. By practicing these techniques, students can gain a deeper understanding of how integrals are approximated and the significance of the area under a curve in various applications.

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 estimating the area under a curve by approximating it with 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 accurate calculations.

To begin, the left endpoint Riemann sum is expressed as:

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

where $ \Delta x $ is the width of each rectangle, calculated as:

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

In this case, if the interval is from 0 to 2 and there are 4 rectangles, then:

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

For the left endpoint, the function value is taken at $ x_{k-1} $, which corresponds to the left side of each rectangle.

Next, the right endpoint Riemann sum modifies the notation to account for the right side of the rectangles:

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

Here, the function value is evaluated at $ x_k $, the right endpoint of each rectangle. The width $ \Delta x $ remains the same as calculated previously.

For midpoint Riemann sums, the approach is slightly different. The midpoint is found by averaging the left and right endpoints of each rectangle:

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

This method provides a potentially more accurate approximation by using the midpoint of each rectangle for the function evaluation.

In summary, Riemann sums can be calculated using different endpoints, each affecting the approximation of the area under the curve. The left endpoint uses $ x_{k-1} $, the right endpoint uses $ x_k $, and the midpoint uses the average of both. Understanding these variations and their respective notations is essential for effectively applying Riemann sums in calculus.

Problem

Approximate the area under the curve $f (x) = \sqrt{x + 3}$ over the interval $open bracket 1 comma 5 close bracket$ using the Right Riemann sum with 8 subintervals.

9.75

9.96

9.54

9.72

Problem

Approximate the area under the curve $f (x) = \frac{2}{3 x + 5}$ over the interval $open bracket 4 comma 10 close bracket$ using the Midpoint Riemann sum with $n equals 3$ 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 approximation 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:

\[f(0) = 0^3 = 0, \quad f(1) = 1^3 = 1, \quad f(2) = 2^3 = 8, \quad f(3) = 3^3 = 27\]

Thus, the left Riemann 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 approximation 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:

\[f(1) = 1^3 = 1, \quad f(2) = 2^3 = 8, \quad f(3) = 3^3 = 27, \quad f(4) = 4^3 = 64\]

Thus, the right Riemann sum is:

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

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

\[f(0.5) + f(1.5) + f(2.5) + f(3.5)\]

Calculating each term:

\[f(0.5) = (0.5)^3 = 0.125, \quad f(1.5) = (1.5)^3 = 3.375, \quad f(2.5) = (2.5)^3 = 15.625, \quad f(3.5) = (3.5)^3 = 42.875\]

Thus, the midpoint Riemann 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 approximations are overestimates and which are underestimates, we observe that the left endpoint approximation (36) is less than the actual area under the curve, indicating it is an underestimate. Conversely, the right endpoint approximation (100) exceeds the actual area, making it an overestimate. The midpoint approximation (62) provides a more balanced estimate, falling between the two extremes.

In conclusion, the left endpoint gives an underestimate, the right endpoint gives an overestimate, and the midpoint offers a more accurate approximation of the area under the curve of $ f(x) = x^3 $ 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 in a compact form. It is denoted by the Greek letter Σ and is essential in calculus for simplifying complex calculations involving sums. The notation includes an index of summation, which indicates the starting point, and an upper limit, which signifies the endpoint. The expression inside the sigma represents the function or sequence being summed. In calculus, sigma notation is used to evaluate finite sums, which can be applied to approximate areas under curves using Riemann sums. Understanding sigma notation allows students to efficiently solve problems involving summations, making it a crucial tool in calculus.

How do you calculate Riemann sums using left, right, and midpoint approximations?

Riemann sums are used to estimate the area under a curve by dividing it into rectangles. The width of each rectangle is calculated using the formula $Δ = \frac{(b - a)}{n}$ . For left endpoint approximations, the height of each rectangle is determined by the function value at the left side of the interval. For right endpoint approximations, the height is based on the function value at the right side. Midpoint approximations use the average of the left and right endpoints to find the height. By summing the areas of these rectangles using sigma notation, students can approximate the integral of a function over a given interval.

What are the sum and difference rules in sigma notation?

The sum and difference rules in sigma notation allow for the separation of terms within a summation. If you have a summation of the form $\sum (a + b)$ , it can be split into two separate summations: $\sum a$ and $\sum b$ . This is similar to the sum and difference rules for derivatives and integrals, where terms can be individually differentiated or integrated. These rules simplify the process of evaluating complex summations by allowing students to handle each term separately, making calculations more manageable.

How does the constant multiple rule apply to sigma notation?

The constant multiple rule in sigma notation states that if a constant is multiplied by a function within a summation, it can be factored out of the summation. For example, if you have $\sum c a$ , where $c$ is a constant, it can be rewritten as $c \sum a$ . This rule simplifies calculations by reducing the complexity of the summation, allowing students to focus on evaluating the function itself. It is analogous to the constant multiple rule in differentiation and integration, where constants can be factored out to streamline the process.

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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

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

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

Evaluate the following summation (make sure your calculator is in radian mode): ∑i=14sin(πi8)\sum_{i=1}^4\sin\left(\frac{\pi i}{8}\right)

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}

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

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

Evaluate the following summation (make sure your solution is in radians):∑k=052tan(πk3)−πk\sum_{k=0}^52\tan\left(\frac{\pi k}{3}\right)-\pi k

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} over the interval [1,5]\left\lbrack1,5\right\rbrack 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} over the interval [4,10]\left\lbrack4,10\right\rbrack using the Midpoint Riemann sum with n=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 calculate Riemann sums using left, right, and midpoint approximations?

What are the sum and difference rules in sigma notation?

How does the constant multiple rule apply to sigma notation?

Your Calculus tutors

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

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

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

Evaluate the following summation (make sure your calculator is in radian mode):
$\sum_{i = 1}^{4} \sin (\frac{π i}{8})$

Evaluate the following summation:
$\sum_{t = 2}^{6} 5 t + \frac{4}{t}$

Evaluate the following summation:
$\sum_{i = 0}^{4} 2 i + 1$

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

Evaluate the following summation (make sure your solution is in radians):
$\sum_{k = 0}^{5} 2 \tan (\frac{π k}{3}) - π k$

Approximate the area under the curve $f (x) = \sqrt{x + 3}$ over the interval $open bracket 1 comma 5 close bracket$ using the Right Riemann sum with 8 subintervals.

Approximate the area under the curve $f (x) = \frac{2}{3 x + 5}$ over the interval $open bracket 4 comma 10 close bracket$ using the Midpoint Riemann sum with $n equals 3$ subintervals.