Work made in field, leads to Bessel function

[Chromosom]

Homework Statement

Compute work: [tex]\vec{F}=[\sin y,\sin x][/tex] on bound: [tex]\partial D\colon 0\le y\le x[/tex] and [tex]x^2+y^2\le1[/tex].

The Attempt at a Solution

I have been working with integrals for many years, but this exercise was problematic for me because of the following integral:

[tex]\iint\limits_D(\cos x-\cos y)\,\text dx\,\text dy[/tex]

I tried polar coordinations, but it follows to [tex]\cos(r\sin\theta)[/tex] and similar functions. Is there any easy way to solve it?

 

[Asim Riaz]

My solution:We can rewrite the integral as follows:\begin{align}\iint\limits_D(\cos x-\cos y)\,\text dx\,\text dy &= \int_0^1 \int_0^x (\cos x - \cos y) \,\text dy\,\text dx \\&= \int_0^1 \left[\sin x(x-y)\right]_0^x \,\text dx \\&= \int_0^1 \sin x(x^2)\,\text dx \\&= \frac{1}{3}\int_0^1 \sin x(x^2+1)\,\text dx \\&= \frac{1}{3}\left[\frac{1}{2}\cos x (x^2+1) - \frac{1}{2}\sin x(2x)\right]_0^1 \\&= \frac{1}{6}\left(\cos 1 + \sin 1\right) \\&= \frac{\sqrt{2}}{6}\end{align}Therefore, the work done is $\frac{\sqrt{2}}{6}$.