Show that ∫a^b(∫x^b f(t) d t) d x=∫a^b(x-a) f(x) d x |...

Name: Problem 65, Chapter 8: Calculus
Uploaded: 2020-04-17T12:06:50-08:00
Duration: 5 min 8 s
Channel: Vikash Ranjan
Description: Show that ∫a^b(∫x^b f(t) d t) d x=∫a^b(x-a) f(x) d x

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Key Concepts

Iterated Integration

Iterated integration involves evaluating a multiple integral by integrating one variable at a time. In this context, a double integral is set up where one integral is nested inside the other. Understanding iterated integration is key to decomposing complex integrals into successive one-dimensional integrals, which can simplify the evaluation process when the integrand or the integration limits depend on one of the variables.

Changing the Order of Integration (Fubini's Theorem)

Changing the order of integration is a technique that relies on Fubini's Theorem, which allows for the interchange of the order of integration in a multiple integral when the function is integrable. By swapping the order of integration, one can often simplify the computation by choosing integration limits that are easier to work with or by exposing a clear structure in the integrals. This concept is particularly helpful in problems where the bounds of one variable depend on another.

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