Iterated Integrals
Iterated Integrals - Related Videos
01:28
Iterated Integrals and Fubini's Theorem
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2026
A double integral generalizes the concept of a single-variable integral to functions of two variables, enabling the computation of the volume beneath a surface z = f(x, y) over a planar region R . For a rectangular region defined by a ≤ x ≤ b and c ≤ y ≤ d, and for functions continuous on this domain, the double integral can be evaluated as an iterated integral. This approach simplifies computation by reducing the problem to successive integrations with respect to one variable at a...
12:38
Structure of HIV-1 Capsid Assemblies by Cryo-electron Microscopy and Iterative Helical Real-space Reconstruction
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Cited by 8 •
2011
This article describes a method to obtain a three-dimensional (3D) structure of helically assembled molecules using cryo-electron microscopy. In this protocol, we use HIV-1 capsid assemblies to illustrate the detailed 3D reconstruction procedure for achieving a density map by the iterative helical real-space reconstruction method.
10:28
Reusable Single Cell for Iterative Epigenomic Analyses
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2022
The present protocol describes a single-cell method for iterative epigenomic analyses using a reusable single cell. The reusable single cell allows analyses of multiple epigenetic marks in the same single cell and statistical validation of the results.
11:42
Iterative Optimization of DNA Duplexes for Crystallization of SeqA-DNA Complexes
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2012
Crystal structure of protein–DNA complexes can provide insight into protein function, mechanism, as well as, the nature of the specific interaction. Here, we report how to optimize the length, sequence and ends of duplex DNA for co-crystallization with Escherichia coli SeqA, a negative regulator of replication initiation.
01:26
Integration by Parts: Indefinite Integrals
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2026
Integration by parts is a fundamental technique in calculus for evaluating integrals involving the product of two functions. It is particularly useful when direct integration is not feasible. The method is based on the product rule for differentiation, which states that the derivative of a product equals the derivative of the first function times the second, plus the first function times the derivative of the second. By integrating this identity and rearranging terms, the integration by parts...
FAQs
The two orders organize the same multivariable accumulation differently, but they are interchangeable only when the relevant conditions are satisfied. One order may produce a simpler inner antiderivative or easier limits than the other. Checking the integration region and the assumptions behind Fubini’s theorem therefore comes before replacing one order with another.
After the first integration, the result generally retains dependence on variables that have not yet been integrated. This intermediate expression can show how accumulation changes across the remaining domain and determines the next calculation. The separation is especially useful in multivariable calculus, where each stage can be handled with respect to one coordinate at a time.
Fubini’s theorem provides the mathematical basis for treating a multiple integral through successive integrations and, when its conditions hold, for changing the order of integration. It distinguishes a valid rearrangement from a merely formal one. In practice, the theorem connects the computational procedure with the underlying properties of the function and its domain.
First identify the variables, the limits assigned to each integration, and which variable is handled first. Then evaluate the inner integral while preserving any dependence on the remaining variable, followed by the outer integration. Comparing an alternative order can be worthwhile if the original limits or intermediate expression make the calculation unnecessarily complicated.
Successive integration combines contributions across more than one variable, allowing a calculation to represent volume or another quantity accumulated over a multidimensional domain. The limits specify the region being included, while the integrand determines the contribution at each location. This interpretation connects symbolic evaluation with geometric and quantitative meaning in multivariable calculus.
The framework also supports mathematical work involving probability, differential equations, and path-dependent functions. In these settings, successive integrations can organize accumulated effects across variables or along a specified progression. Their broader importance comes from linking a concrete calculation procedure with ideas in mathematical analysis, multivariable calculus, and geometry.