Riemann sums
E47353
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
Aliases (1)
- Darboux sum ×1
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
construction of the definite integral
→
mathematical concept → numerical approximation method → |
| appearsIn |
introductory calculus courses
→
real analysis textbooks → |
| assumes |
boundedness of the function on the interval
→
|
| basedOn |
partition of an interval
→
sum of areas of rectangles → |
| conditionForConvergence |
function being Riemann integrable
→
|
| contrastWith |
Monte Carlo integration
→
Simpson's rule → |
| convergesTo |
value of the Riemann integral when the function is Riemann integrable
→
|
| coreIdea |
approximate area by rectangles over subintervals
→
|
| dependsOn |
choice of partition
→
choice of sample points in each subinterval → |
| domain |
real-valued functions on closed intervals
→
|
| field |
calculus
→
numerical analysis → real analysis → |
| generalization |
Riemann–Stieltjes sums
→
multiple Riemann sums for multivariable integration → |
| hasComponent |
function values at sample points
→
partition points → sample points → subinterval widths → |
| hasNotation |
sum from i equals 1 to n of f(x_i^*) Δx_i
→
|
| hasType |
Darboux sum
→
left Riemann sum → lower Riemann sum → midpoint Riemann sum → right Riemann sum → trapezoidal sum → upper Riemann sum → |
| introducedIn |
19th century
→
|
| limitDefinition |
definite integral as limit of Riemann sums
→
|
| namedAfter |
Bernhard Riemann
→
|
| prerequisiteFor |
understanding Riemann integration theory
→
|
| refinementProperty |
finer partitions generally give better approximations
→
|
| relatedTo |
Darboux integral
→
Lebesgue integral → Riemann integral → |
| sufficientConditionForIntegrability |
function being bounded and having only finitely many discontinuities
→
function being continuous on a closed interval → |
| usedFor |
approximating definite integrals
→
approximating the area under a curve → defining the definite integral → |
| usedIn |
error estimation for integrals
→
numerical integration → rigorous proofs of the Fundamental Theorem of Calculus → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
Riemann sums
("Darboux sum")
→
|
hasType |
|
Bernhard Riemann
→
|
knownFor |
|
Friedrich Bernhard Riemann
→
|
notableConcept |