MathematicsDefinite integrals as Riemann sumsUniversity

IBUndergraduate

Faulhaber Formulas

Lists standard finite-sum formulas used when evaluating Riemann sums.

Understand the formulaSee the free derivationOpen the full walkthrough

i = 1 \sum n i = \frac{n ( n + 1 )}{2}; i = 1 \sum n i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}; i = 1 \sum n i^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Open Full Walkthrough Try Calculator

This public page keeps the free explanation visible and leaves premium worked solving, advanced walkthroughs, and saved study tools inside the app.

Core idea

Overview

Lists standard finite-sum formulas used when evaluating Riemann sums. It explains how the rule changes or interprets an integral, including the conditions that make the shortcut valid. The main goal is to help students set up the expression correctly before doing any algebra or calculation.

When to use: Use this when the problem matches the stated limit, antiderivative, summation, or definite-integral pattern.

Why it matters: These rules connect limits, sums, and antiderivatives to practical integral calculations.

Symbols

Variables

result = result

result

Variable

Walkthrough

Derivation

Derivation of Sums of powers of positive integers

Lists standard finite-sum formulas used when evaluating Riemann sums.

n is a positive integer.
The index i runs from 1 to n.

State the verified result

This is the standard calculus statement for the entry.

i = 1 \sum n i = \frac{n ( n + 1 )}{2}; i = 1 \sum n i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}; i = 1 \sum n i^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Check the conditions

The conclusion is valid only under the listed assumptions.

i = 1 \sum n i = \frac{n ( n + 1 )}{2}; i = 1 \sum n i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}; i = 1 \sum n i^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Result

i = 1 \sum n i = \frac{n ( n + 1 )}{2}; i = 1 \sum n i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}; i = 1 \sum n i^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Source: OpenStax, Calculus Volume 1, Section 5.2: The Definite Integral, accessed 2026-04-09

Free formulas

Rearrangements

Solve for $\sum_{i = 1}^{n} i = \frac{n ( n + 1 )}{2}; \sum_{i = 1}^{n} i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}; \sum_{i = 1}^{n} i^{3} = [\frac{n ( n + 1 )}{2}]^{2}$

Use Sums of powers of positive integers

i = 1 \sum n i = \frac{n ( n + 1 )}{2}; i = 1 \sum n i^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}; i = 1 \sum n i^{3} = [\frac{n ( n + 1 )}{2}]^{2}

Check the conditions and apply the stated rule; this concept-only entry has no algebraic solver rearrangement.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

Contains advanced operator notation (integrals/sums/limits)

Why it behaves this way

Intuition

Limits and integrals are controlled by structure: quotient forms compare rates, indefinite integrals reverse differentiation, and Riemann sums build area from many thin pieces.

Σ

summation

Adds indexed terms.

n

upper index

The number of terms or partitions.

i

index

The running counter in the sum.

Signs and relationships

+C: Indefinite integrals represent a family because constants differentiate to zero.
-: Reversing definite-integral bounds reverses interval orientation.

Free study cues

Insight

Canonical usage

These formulas are used to calculate the sum of powers of integers, which are fundamental in approximating definite integrals using Riemann sums.

Common confusion

Students may sometimes try to assign units to 'n' or the summation result, forgetting that these formulas deal with abstract counts of terms rather than physical quantities.

Dimension note

The Faulhaber formulas themselves produce dimensionless integer or rational number results, as they sum dimensionless integers. When used in the context of Riemann sums, these sums represent a count of subintervals or a

Unit systems

$n$ dimensionless · Represents the number of terms in the summation, which is always a dimensionless integer.

$i$ dimensionless · The index of summation, which iterates through dimensionless integers.

One free problem

Practice Problem

What is sum i from 1 to n?

out0

Solve for: result

Hint: Check the form and required conditions first.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Area, accumulation, and limiting processes in physics and engineering are modeled with these integral and limit rules.

Study smarter

Tips

Check the condition before applying the rule.
Include +C for indefinite integrals.
Replace scraped infinity fragments with proper infty notation.

Avoid these traps

Common Mistakes

Using the rule without checking its form or hypothesis.
Forgetting the constant of integration or the sign change from reversed bounds.

Common questions

Frequently Asked Questions

Lists standard finite-sum formulas used when evaluating Riemann sums.

Use this when the problem matches the stated limit, antiderivative, summation, or definite-integral pattern.

These rules connect limits, sums, and antiderivatives to practical integral calculations.

Using the rule without checking its form or hypothesis. Forgetting the constant of integration or the sign change from reversed bounds.

Area, accumulation, and limiting processes in physics and engineering are modeled with these integral and limit rules.

Check the condition before applying the rule. Include +C for indefinite integrals. Replace scraped infinity fragments with proper infty notation.

References

Sources

OpenStax, Calculus Volume 1, Section 5.2: The Definite Integral, accessed 2026-04-09
Wikipedia: Summation, accessed 2026-04-09
Wikipedia:Faulhaber's formula
Wikipedia:Riemann sum
Wolfram MathWorld - Faulhaber's Formula
Wikipedia - Faulhaber's formula

Faulhaber Formulas

Overview

Variables

Derivation

State the verified result

Check the conditions

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Area as a Riemann sum

Linearity Properties of Summation

Frequently Asked Questions

Sources