Evaluating Double Integrals: A Symmetric Case Study

To evaluate the double integral

I (displaystyle int_0^2 int_0^2 e^{-t-u} , dt , du)

Introduction

Double integrals are a fundamental concept in multivariable calculus, used to calculate areas, volumes, and other quantities in two dimensions. This article presents a detailed solution to a specific double integral, focusing on the application of symmetry properties to simplify the calculation.

Problem Statement

We are asked to evaluate the double integral

I (displaystyle int_0^2 int_0^2 e^{-t-u} , dt , du)

To solve this, we will leverage the fact that the region of integration and the integrand have a symmetric property about the line u t.

Utilizing Symmetry

The region of integration is a square in the tu-plane, and the function e-t-u is symmetric about the line u t. Therefore, we can simplify our calculation by taking advantage of this symmetry.

By considering the symmetry, we can rewrite the integral as

I 2 (displaystyle int_0^2 int_0^t e^{-t-u} , du , dt)

Since the integral is symmetric, we can multiply the inner integral by 2 and rewrite the outer integral from 0 to t.

Evaluation of the Integral

Now, we will perform the inner integral first:

2 (displaystyle int_0^2 -e^{-t-u} Big|_{u0}^{ut} , dt 2 int_0^2 e^{-t} , dt)

After substituting the limits and simplifying, we get:

2 (displaystyle int_0^2 e^{-t} , dt -2e^{-t} Big|_0^2 -2(e^{-2} - e^0) -2e^{-2} 2 2 - 2e^{-2})

Therefore, we can express the final result as

I 2(1 - e-2)

Conclusion

Utilizing the symmetry of the integrand and region of integration allowed us to simplify the evaluation of the double integral. This approach not only makes the calculation more straightforward but also demonstrates the power of symmetry in simplifying complex problems in multivariable calculus.

Related Articles

Integration Techniques in Calculus Applications of Symmetry in Mathematics Solving Differential Equations Using Symmetry

Further Reading

To explore more about double integrals, integration techniques, and symmetrical properties, consider the following resources:

Introduction to Double Integrals: Symmetry and Beyond Symmetry in Calculus: A Comprehensive Guide Using Online Calculators for Evaluating Double Integrals