Questions: Use symmetry to evaluate the double integral. ∫∫R x^13 dA, R=[-4,4] × [0,5]

Solution Steps

To evaluate the double integral \(\iint_{\mathcal{R}} x^{13} dA\) over the region \(\mathcal{R} = [-4, 4] \times [0, 5]\), we can use the property of symmetry. The integrand \(x^{13}\) is an odd function with respect to \(x\), and the region \(\mathcal{R}\) is symmetric about the \(y\)-axis. Therefore, the integral of an odd function over a symmetric interval around zero is zero.

We need to evaluate the double integral

\[ \iint_{\mathcal{R}} x^{13} \, dA \]

over the region \(\mathcal{R} = [-4, 4] \times [0, 5]\).

The integrand \(x^{13}\) is an odd function because it satisfies the property \(f(-x) = -f(x)\). Specifically, for any \(x\), we have:

\[ (-x)^{13} = -x^{13} \]

The region \(\mathcal{R}\) is symmetric about the \(y\)-axis. This means that for every point \((x, y)\) in the region, the point \((-x, y)\) is also in the region.

Since the integrand is odd and the region of integration is symmetric about the \(y\)-axis, the integral evaluates to zero:

\[ \iint_{\mathcal{R}} x^{13} \, dA = 0 \]

Final Answer