Lower sum calculator - Take A=1 until Z=26, and add the numeric values found for each letter (the letters calculator sum up automatically) to get the word's value: Example: DCODE = D+C+O+D+E = 4+3+15+4+5 = 31. Numerology is a pseudo-science, it is a belief without scientific foundation, attention to its use by charlatans. Alphabet letters default value chart:

 
Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Simpson's Rule. Save Copy. Log InorSign Up. Simpson's rule for definite integrals: Enter a function f(x), use the a and b sliders to choose the limits of integration, and use the n .... Used cluster mailboxes for sale

Jan 7, 2020 · UPPER AND LOWER SUM EXAMPLES MATH 15200, SECTION 13 This just contains some relatively simple examples of computing upper and lower sums, to get an idea. First, recall the de nitions: if [a;b] is a closed interval, then P = fx ign i=0 is a partition of [a;b] if a = x 0 < x 1 < x 2 < < x n 1 < x n = b. We let x i = x i x i 1. Simply, there is an interval [a,b] called the limits, bounds or boundaries. This type can be defined as the limit of the integral sums when the diameter of partitioning tends to zero. Our online definite integral calculator with bounds evaluates the integrals by considering the upper and lower limit of the function.More control over their money. When you take a lump sum, you can make investment and distribution decisions that a pension does not offer. Ability to leave money to beneficiaries. With a lump sum, you can withdraw money to cover your retirement income needs and leave whatever is left to your beneficiaries. Fear that pension will collapse.Regression Sum of Squares Calculator Total Sum of Squares Calculator ... Upper and Lower Fence Calculator Attributable Risk Calculator Number Needed to Harm Calculator Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this siteLimit calculator with steps shows the step-by-step solution of limits along with a plot and series expansion. It employs all limit rules such as sum, product, quotient, and L'hopital's rule to calculate the exact value. You can evaluate limits with respect to \(\text{x , y, z , v, u, t}\) and \(w\) using this limits calculator. That’s not it.Free Minimum Calculator - find the Minimum of a data set step-by-step We have ... Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal ... {\sum\space\int\space\product} …The logarithm calculator simplifies the given logarithmic expression by using the laws of logarithms. Step 2: Click the blue arrow to submit. Choose "Simplify/Condense" from the topic selector and click to see the result in our Algebra Calculator! Examples. Simplify/Condense Simplify/Condense Simplify/Condense Simplify/Condense . Popular Problems Regression Sum of Squares Calculator Total Sum of Squares Calculator ... Upper and Lower Fence Calculator Attributable Risk Calculator Number Needed to Harm CalculatorSimply, there is an interval [a,b] called the limits, bounds or boundaries. This type can be defined as the limit of the integral sums when the diameter of partitioning tends to zero. Our online definite integral calculator with bounds evaluates the integrals by considering the upper and lower limit of the function.Calculator for adding and subtracting fractions with like or unlike denominators. The fraction calculator can add or subtract 2 fractions, 3 fractions and up to 9 fractions at a time, and shows the work to find common denominators, and simplify fractions to lowest terms or mixed number answers.Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.Partition Calculator. Super simple, just gives you the number of partitions and other relevant data. Get the free "Partition Calculator" widget for your website, blog, Wordpress, Blogger, or iGoogle. Find more Mathematics widgets in Wolfram|Alpha.This calculus video tutorial provides a basic introduction into summation formulas and sigma notation. It explains how to find the sum using summation formu...Arrange data points from smallest to largest and locate the central number. This is the median. If there are 2 numbers in the middle, the median is the average of those 2 numbers. The mode is the number in a data set that occurs most frequently. Count how many times each number occurs in the data set. The mode is the number with the highest tally.Riemann Sums - Rectangular Approximation (LRAM, RRAM, MRAM) Activity. Ισίδωρος Γλαβάς. Estimate the Sum Calculator by Rounding to Nearest 10, 100 or 1000. Estimate the sum calculator is a pre-algebra tool to find the actual and estimated sum for the given addends by rounding off to the nearest ten, hundred and thousand. Just supply the addends, this calculator rounds the addends to the nearest 10, 100 and 1000 and performs the ...In the unlikely case you are actually asked to evaluate with different Δx values, you would simply take each value of Δx and multiply it with its corresponding f (x) value and add them all together. For example, if you had a table that listed several x values such as 1, 3, 7 and 10 as well as their respective f (x) values, say, 6, 7, 3 and 5 ...This normal distribution calculator (also a bell curve calculator) calculates the area under a bell curve and establishes the probability of a value being higher or lower than any arbitrary value X. You can also use this probability distribution calculator to find the probability that your variable is in any arbitrary range, X to X₂, just by ...Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Dr Adrian Jannetta Topic: Area, Upper and Lower Sum or Riemann Sum This applet shows how upper and lower Riemann sums can approximate an integral Further, they show that as the number of strips increases, the Riemann sums converge to true value of the definite integral. With help of this calculator you can: find the matrix determinant, the rank, raise the matrix to a power, find the sum and the multiplication of matrices, calculate the inverse matrix. Just type matrix elements and click the button. Leave extra cells empty to enter non-square matrices. You can use decimal fractions or mathematical expressions: The upper bound is the value that helps us sum integral at its maximum value. The upper bound is denoted as U, and its determination is crucial in the integration process. You can enter the upper bound of your limit in the upper bound section of the upper bound calculator. Step 4: Give Lower bound valueYou decide to make an additional $300 payment toward principal every month to pay off your home faster. By adding $300 to your monthly payment, you’ll save just over $64,000 in interest and pay off your home over 11 years sooner. Consider another example. You have a remaining balance of $350,000 on your current home on a 30-year fixed rate ...Free power sums calculator - calculate power sums step-by-step ... Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile ... A sum of series, a.k.a. summation of sequences is adding up all values in an ordered series, usually expressed in sigma (Σ) notation. A series can be finite or infinite depending on the limit values. Using the summation calculator. In "Simple sum" mode our summation calculator will easily calculate the sum of any numbers you input. You can ...The procedure to use the Riemann sum calculator is as follows: Step 1: Enter the function, upper and lower bound in the input field. Step 2: Now click the button “Submit” to get the Riemann sum. Step 3: Finally, the Riemann sum using upper and lower bound will be displayed in the new window. The Riemann Sum Calculator works by using the following formula: ( 1) ∫ a b f ( x) d x ≈ S = ∑ k = 1 n f ( x = x k) ( Δ x) A curve defined by f (x) over a closed interval [a, b] can be split into n rectangles (sub-intervals) each of length b − a n with endpoints [i k, f k ]. The height of the kth rectangle then equals the value of the ... The logarithm calculator simplifies the given logarithmic expression by using the laws of logarithms. Step 2: Click the blue arrow to submit. Choose "Simplify/Condense" from the topic selector and click to see the result in our Algebra Calculator! Examples. Simplify/Condense Simplify/Condense Simplify/Condense Simplify/Condense . Popular Problems The Riemann Sum Calculator works by using the following formula: ( 1) ∫ a b f ( x) d x ≈ S = ∑ k = 1 n f ( x = x k) ( Δ x) A curve defined by f (x) over a closed interval [a, b] can be split into n rectangles (sub-intervals) each of length b − a n with endpoints [i k, f k ]. The height of the kth rectangle then equals the value of the ... Jul 20, 2020 · Calculator for adding and subtracting fractions with like or unlike denominators. The fraction calculator can add or subtract 2 fractions, 3 fractions and up to 9 fractions at a time, and shows the work to find common denominators, and simplify fractions to lowest terms or mixed number answers. The definite integral of a continuous function f over the interval [ a, b] , denoted by ∫ a b f ( x) d x , is the limit of a Riemann sum as the number of subdivisions approaches infinity. That is, ∫ a b f ( x) d x = lim n → ∞ ∑ i = 1 n Δ x ⋅ f ( x i) where Δ x = b − a n and x i = a + Δ x ⋅ i .An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed number to the previous term. It is represented by the formula a_n = a_1 + (n-1)d, where a_1 is the first term of the sequence, a_n is the nth term of the sequence, and d is the common difference, which is obtained by subtracting the previous term ...An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed number to the previous term. It is represented by the formula a_n = a_1 + (n-1)d, where a_1 is the first term of the sequence, a_n is the nth term of the sequence, and d is the common difference, which is obtained by subtracting the previous term ...Oct 10, 2023 · For a given function f (x) over a partition of a given interval, the lower sum is the sum of box areas m^*Deltax_k using the infimum m of the function f (x) in each subinterval [x_ (k-1),x_k]. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed number to the previous term. It is represented by the formula a_n = a_1 + (n-1)d, where a_1 is the first term of the sequence, a_n is the nth term of the sequence, and d is the common difference, which is obtained by subtracting the previous term ...Input: First, select a calculation method either the simple sum or sigma notation sum. If you selected a simple sum, then enter numbers or series separated with a comma. When selecting the sigma notation, then enter an equation with start and end value. Hit the calculate button to see the summation of a constant and numbers. Let be an arbitrary point in the th subinterval. Then the quantity. is called a Riemann sum for a given function and partition, and the value is called the mesh size of the partition. If the limit of the Riemann sums exists as , this limit is known as the Riemann integral of over the interval . The shaded areas in the above plots show the lower ...We can square n each time and sum the result: 4. ... You can try some of your own with the Sigma Calculator. Why is it called "Sigma" Sigma is the upper case letter S ... Gives the approximation of a definite integral using the Trapezoidal Rule. Send feedback | Visit Wolfram|Alpha. Function. Number of Trapezoids. Lower Limit. Upper Limit. Submit. Get the free "Trapezoidal Rule Calculator" widget …The Riemann Sum Calculator works by using the following formula: ( 1) ∫ a b f ( x) d x ≈ S = ∑ k = 1 n f ( x = x k) ( Δ x) A curve defined by f (x) over a closed interval [a, b] can be split into n rectangles (sub-intervals) each of length b − a n with endpoints [i k, f k ]. The height of the kth rectangle then equals the value of the ...Array formulas must be entered with Ctrl + Shift + Enter rather than just the Enter key. This answer will return #REF!, if i = 0. You don’t need complexe functions, Just calculate (max-min+1)* ( (formula min)+ (formula max))/2. For the average you can use the function AVG but this is the no functions solution.Take A=1 until Z=26, and add the numeric values found for each letter (the letters calculator sum up automatically) to get the word's value: Example: DCODE = D+C+O+D+E = 4+3+15+4+5 = 31. Numerology is a pseudo-science, it is a belief without scientific foundation, attention to its use by charlatans. Alphabet letters default value chart:Calculus plays a fundamental role in modern science and technology. It helps you understand patterns, predict changes, and formulate equations for complex phenomena in fields ranging from physics and engineering to biology and economics. Essentially, calculus provides tools to understand and describe the dynamic nature of the world around us ...Traverse the matrix and calculate the sum for upper and lower triangles accordingly. Time Complexity: O (r * c), Where r and c represent the number of rows and columns of the given matrix. Auxiliary Space: O (1), No extra space is required, so it is a constant. This article is contributed by Pranav.Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.A maxterm is an expression grouping Boolean variables, complemented or not (a or not (a)), linked by logical ORs and with a value of 0. The maxterm is described as a sum of product (SOP). Example: a AND b AND c = 0 or (NOT (a) OR b) AND (NOT (c) OR d) = 0 are maxterms. Each row of a logical truth table worth 0/False can therefore be associated ...Definition. The definition of the Darboux integral considers upper and lower (Darboux) integrals, which exist for any bounded real -valued function on the interval The Darboux integral exists if and only if the upper and lower integrals are equal. The upper and lower integrals are in turn the infimum and supremum, respectively, of upper and ...To calculate the Left Riemann Sum, utilize the following equations: 1.) A r e a = Δ x [ f ( a) + f ( a + Δ x) + f ( a + 2 Δ x) + ⋯ + f ( b − Δ x)] 2.) Δ x = b − a n. Where Δ x is the length of each subinterval (rectangle width), a is the left endpoint of the interval, b is the right endpoint of the interval, and n is the desired ...Even if you don’t have a physical calculator at home, there are plenty of resources available online. Here are some of the best online calculators available for a variety of uses, whether it be for math class or business.Simpson’s Rule Example 1. Let's say we want to find the area under the curve of the function f (x)=x^2 f (x) = x2 from x=0 x = 0 to x=2 x = 2 using Simpson's Rule. We divide the interval [0,2] [0,2] into 4 4 equal subintervals, so each subinterval has a width of h=\frac {2-0} {4}= 0.5 h = 42−0 = 0.5. Now we evaluate the function at each ...Limits Calculator online with solution and steps. Detailed step by step solutions to your Limits problems with our math solver and online calculator. ... The derivative of a sum of two or more functions is the sum of the derivatives of each function $\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-4\right)$Features: High Precision: Calculate the sum with up to 1000 decimal precision. User-Friendly: No need to use the plus (+) sign between the numbers. Supports Large Numbers: Easily add numbers with thousand separators. Mobile Version Available: Install it as a PWA on your home screen and add up numbers on the go.The total area of the inscribed rectangles is the lower sum, and the total area of the circumscribed rectangles is the upper sum . By taking more rectangles, you get a better approximation. In the limit, as the number of rectangles increases "to infinity", the upper and lower sums converge to a single value, which is the area under the curve .Given, two numbers are 567 and 349. First, round up the first number to the nearest 10's place value ie., 567 to 570. 570+349. Now, round up the second number to the nearest 10's place value ie., 349 to 350. 570+350. Add 570 and 350. = 920. 567+349≈920. ∴ The Estimated Sum of 567+349 is 920.Free Simpson's Rule calculator - approximate the area of a curve using Simpson's rule step-by-step ... Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal ...This sigma sum calculator computes the sum of a series over a given interval. Fill in the variables 'from', 'to', type an expression then click on the button calculate. This summation notation calculator can sum up many types of sequencies including the well known arithmetic and geometric sequencies, so it can help you to find the terms ...Limits at infinity are used to describe the behavior of a function as the input to the function becomes very large. Specifically, the limit at infinity of a function f (x) is the value that the function approaches as x becomes very large (positive infinity). what is a one-sided limit?This calculus video tutorial provides a basic introduction into summation formulas and sigma notation. It explains how to find the sum using summation formu...Upper and Lower Sum or Riemann Sum. Parent topic: Integral Calculus. Integral Calculus Math Upper/Lower Sum. Definite Integral Illustrator (I) Activity. ... Riemann Sum Calculator. Activity. megan.ann.martinez. Integral. Activity. mtamim1. Riemann Sums - Rectangular Approximation (LRAM, RRAM, MRAM)Oct 10, 2023 · The Darboux integral, also called a Darboux-Stieltjes integral, is a variant of the Stieltjes integral that is defined as a common value for the lower and upper Darboux integrals. Let and be bounded real functions on an interval , with nondecreasing. For any partition given by , let . The lower Darboux integral is the supremum of all lower sums ... The procedure to use the Riemann sum calculator is as follows: Step 1: Enter the function, upper and lower bound in the input field. Step 2: Now click the button “Submit” to get the Riemann sum. Step 3: Finally, the Riemann sum using upper and lower bound will be displayed in the new window. The calculator allows to calculate the terms of an arithmetic sequence between two indices of this sequence. Thus, to obtain the terms of an arithmetic sequence defined by un = 3 + 5 ⋅ n u n = 3 + 5 ⋅ n between 1 and 4 , enter : sequence ( 3 + 5 ⋅ n; 1; 4; n 3 + 5 ⋅ n; 1; 4; n) after calculation, the result is returned.The "complete" gamma function Gamma(a) can be generalized to the incomplete gamma function Gamma(a,x) such that Gamma(a)=Gamma(a,0). This "upper" incomplete gamma function is given by Gamma(a,x)=int_x^inftyt^(a-1)e^(-t)dt. (1) For a an integer n Gamma(n,x) = (n-1)!e^(-x)sum_(k=0)^(n-1)(x^k)/(k!) (2) = (n-1)!e^(-x)e_(n-1)(x), …The low points of the curve coincide with the left edges of the rectangles, at the points (2, 12) and (3, 27). Therefore the areas of the rectangles are 1×12 = 12 and 1×27 = 27, and the total or lower sum is S (2) = 12+27 = 39. (The lower sum is written with a lower-case s to distinguish it from the upper sum’s upper-case S.)Additional lump sum payment. R. Calculate. Current values. Projected values ... reducing your capital. This has the effect of reducing your loan term, thus ...Regression Sum of Squares Calculator Total Sum of Squares Calculator ... Upper and Lower Fence Calculator Attributable Risk Calculator Number Needed to Harm Calculator In mathematics, a Riemann sum is a type of estimation of a definite integral by a finite sum with a specified lower and upper bound, which was founded by B.Riemann (1826−1866), a German mathematician. Generally, the Riemann sum is used to determine the integration process and it is a systematic way to calculate the curved surface area.The Octal Calculator is used to perform addition, subtraction, multiplication and division on two octal numbers (Step by Step). Octal In mathematics and computer science, octal (oct for short) is a positional numeral system with a base of 8, and uses the digits 0 to 7.Symbolab is the best calculus calculator solving derivatives, integrals, limits, series, ODEs, and more. What is differential calculus? Differential calculus is a branch of calculus that includes the study of rates of change and slopes of functions and involves the concept of a derivative.However, this isn't necessary, since you can just change the expression inside the sum to get the same effect. So if you have the sum from i=1 to 10 of i², and you want to i to step by 1/3 instead of 1, you can change the expression i² to (1+(i-1)/3)², and have i …How to use the summation calculator. Input the expression of the sum. Input the upper and lower limits. Provide the details of the variable used in the expression. Generate the results by clicking on the "Calculate" button. Summation (Sigma, ∑) Notation Calculator. k =. Left and right Riemann sums. To make a Riemann sum, we must choose how we're going to make our rectangles. One possible choice is to make our rectangles touch the curve with their top-left corners. This is called a left Riemann sum. The shaded area below the curve is divided into 4 rectangles of equal width. Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Estimating Area Under a Curve. Save Copy ... 1. f x = 2. Let a = lower bound of your interval and let b = upper bound of your interval 3. a = − 1 0. 4. b = − 1 ...Limits at infinity are used to describe the behavior of a function as the input to the function becomes very large. Specifically, the limit at infinity of a function f (x) is the value that the function approaches as x becomes very large (positive infinity). what is a one-sided limit? Examples: Apply 10% to 100, and see how each value was calculated. Try 12.5% of 50: the "50 less 12.5%" value is 43.75. Then try 12.5% of 43.75 (shows that before a 12.5% reduction it was 50.) Compare 100 to 110, and see that to go from 100 to 110 is a 10% increase, but to go from 110 back down to 100 is a 9.09% decrease (not a 10% decrease)The Darboux integral, also called a Darboux-Stieltjes integral, is a variant of the Stieltjes integral that is defined as a common value for the lower and upper Darboux integrals. Let and be bounded real functions on an interval , with nondecreasing. For any partition given by , let . The lower Darboux integral is the supremum of all lower sums ...Free log equation calculator - solve log equations step-by-step We have updated our ... Mean Geometric Mean Quadratic Mean Median Mode Order Minimum Maximum Probability Mid-Range Range Standard Deviation Variance Lower Quartile Upper Quartile Interquartile Range Midhinge Standard Normal ... {\sum\space\int\space\product} \bold{\begin{pmatrix ...The logarithm calculator simplifies the given logarithmic expression by using the laws of logarithms. Step 2: Click the blue arrow to submit. Choose "Simplify/Condense" from the topic selector and click to see the result in our Algebra Calculator! Examples. Simplify/Condense Simplify/Condense Simplify/Condense Simplify/Condense . Popular Problems

This is an additional payment that you can make at any time. You can make lump sum payments as often as you like, but if you exceed your overpayment allowance, you’ll have to pay an ERC on the amount over 20%. If you pay £1,000 or more, your monthly payment will be recalculated. If you want to pay off your mortgage or you have less than £ .... Ameris online banking sign in

lower sum calculator

First step is to select the right Riemann sum calculator from the calculator. Select the compute endpoint approximation of right, left, midpoint from the calculator. There are some of the mentioned variables “From x= to Menu. To” on the calculator. You can also manage the upper and lower values. Angles Calculator - find angle, given anglesLVR and interest rates. Why is this important to you? A lower LVR usually translates to a lower interest rate for the borrower.The lower sum L( n[ 1, 1], x2 + 1) computes the area of the green region pictured below. f(x) = x2 + 1 x picture of an upper sum The upper sum U( n[ 1, 1], x2 + 1) computes the area …We can square n each time and sum the result: 4. ... You can try some of your own with the Sigma Calculator. Why is it called "Sigma" Sigma is the upper case letter S ... the lower integral of f over [a, b] and. ¯ ∫b af = inf {U(f, P): P is a partition of [a, b]} the upper integral of f over [a, b]. Note that both the lower integral and the upper integral are finite real numbers since the lower sums are all bounded above by any upper sum and the upper sums are all bounded below by any lower sum.For a more rigorous treatment of Riemann sums, consult your calculus text. The following Exploration allows you to approximate the area under various curves under the interval $[0, 5]$. You can create a partition of the interval and view an upper sum, a lower sum, or another Riemann sum using that partition. The Exploration will give you the ...This is an additional payment that you can make at any time. You can make lump sum payments as often as you like, but if you exceed your overpayment allowance, you’ll have to pay an ERC on the amount over 20%. If you pay £1,000 or more, your monthly payment will be recalculated. If you want to pay off your mortgage or you have less than £ ...This is an online Karnaugh map generator that makes a kmap, shows you how to group the terms, shows the simplified Boolean equation, and draws the circuit for up to 6 variables. It also handles Don't cares. The Quine-McCluskey solver can be used for up to 6 variables if you prefer that. Select the number of variables, then choose SOP (Sum of ...picture a Lower Riemann Sum. Let’s calculate these quantities. Each rectangle is of width 0.2. In the Upper Sum, the height of each rectangle is f evaluated at the right endpoint of the subinterval; in the Lower Sum the heights are fevaluated at the left endpoint of the subinterval). Upper Sum = :2f(:2)+:2f(:4)+:2f(:6)+:2f(:8)+:2f(1) = 8 5What to know about loan repayments. Most loans are installment loans, meaning that you receive a lump sum of money upfront that you pay back through a course of monthly payments.If you have a ...A Riemann sum is an approximation of a region&#x27;s area, obtained by adding up the areas of multiple simplified slices of the region. It is applied in calculus to formalize the method of exhaustion, used to determine the area of a region. This process yields the integral, which computes the value of the area exactly. Let us decompose a given closed …With terms defined as in a double Riemann sum, the double integral of f over R is. ∬ R f ( x, y) d A = lim m, n → ∞ ∑ j = 1 n ∑ i = 1 m f ( x i j ∗, y i j ∗) ⋅ Δ A. 🔗. Some textbooks use the notation ∫ R f ( x, y) d A for a double integral. You will see this in some of the WeBWorK problems. Firstly, if a diagonal element equals zero, this method won't work. Secondly, during the calculation the deviation will rise and the further, the more. So the result won't be precise. For the deviation reduction, the Gauss method modifications are used. They are based on the fact that the larger the denominator the lower the deviation.3/8 = 3 ÷ 8 = .375 inches. Convert any feet to inches by multiplying the number of feet by 12, since there are 12 inches in a foot. Using a foot to inch conversion calculator also simplifies this step. For example: 3 feet = 3 × …We begin our exploration of integration and integrability by looking at the notion of a partition of an interval and the corresponding upper and lower sum of....

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