Integral Calculus inequalities problem

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SUMMARY

The discussion centers on proving the inequality \(\frac{1}{e} \leq \frac{1}{4\pi^{2}} \int_{R} e^{\cos(x-y)} dx dy \leq e\), where \(R\) is defined as the region \([0, 2\pi] \times [0, 2\pi]\). Participants suggest utilizing the Mean Value Theorem and the Intermediate Value Theorem to approach the problem. A key insight involves determining the minimum value of \(e^{\cos(x-y)}\) over the region \(R\) to establish the bounds of the integral.

PREREQUISITES
  • Understanding of integral calculus, specifically double integrals.
  • Familiarity with the Mean Value Theorem and Intermediate Value Theorem.
  • Knowledge of the properties of the exponential function, particularly \(e^{\cos(x)}\).
  • Basic understanding of inequalities in mathematical proofs.
NEXT STEPS
  • Study the properties of the exponential function and its behavior over intervals.
  • Learn about double integrals and how to evaluate them over specified regions.
  • Explore the Mean Value Theorem and its applications in calculus.
  • Investigate techniques for proving inequalities in calculus.
USEFUL FOR

This discussion is beneficial for students studying calculus, particularly those tackling integral inequalities, as well as educators looking for teaching strategies related to advanced calculus concepts.

tomelwood
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Homework Statement


Hey, just wondering how I might go about doing this problem, as I really have very little idea...

Prove the following inequality:
\frac{1}{e}\leq\frac{1}{4\pi^{2}}\int_{R}e^{cos(x-y)}dxdy\leqe
(hopefully this reads "one over e is less than or equal to one over four pi squared times the integral over R of e to the power of cos(x-y) dx dy which is less than or equal to e"

Homework Equations



R is the region [0,2pi]x[0,2pi]

The Attempt at a Solution


I think the Mean value, and intermediate value theorem may come into it somewhere, but I really don't know where to begin. Any pointers at all would be greatly appreciated.
Many thanks
 
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Hint: what is the minimal value of e^{\cos(x-y)} on the whole R?
 

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