Question: "Use symmetry to evaluate the double integral. ∫∫Rxy1+x4dA=∫−11xy1+x4dydx"

vestirme4

Answered question

2021-10-18

Use symmetry to evaluate the double integral.

\int \int_{R} \frac{x y}{1 + x^{4}} d A = \int_{- 1}^{1} \frac{x y}{1 + x^{4}} d y d x

ensojadasH

Skilled2021-10-19Added 100 answers

We are expected to use symmetry to solve this problem

\int \int_{R} \frac{x y}{1 + x^{4}} d A = \int_{- 1}^{1} \int_{0}^{1} \frac{x y}{1 + x^{4}} d y d x

I = \int_{- 1}^{1} \int_{0}^{1} \frac{x y}{1 + x^{4}} d y d x

Substitute

x = - u

And

d x = - d u

\int_{1}^{- 1} \int_{0}^{1} \frac{- (u) y}{1 + {(- u)}^{4}} d y (- d u)

Remember that:

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x

\int_{- 1}^{1} \int_{0}^{1} - \frac{u y}{1 + u^{4}} d y d u

Remember that:

\int_{a}^{b} f (u) d u = \int_{a}^{b} f (x) d x

\int_{- 1}^{1} \int_{0}^{1} - \frac{x y}{1 + x^{4}} d y d x

(2)
Add equation (1) and equation (2), to get

2 I = \int_{- 1}^{1} \int_{0}^{1} 0 d y d x = 0

Result:

\int \int_{R} \frac{x y}{1 + x^{4}} d A = 0

Substitute x=-u
And then add the resultant to the original integral

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