Area of A in (x,y)-plane given by a paramtric equation

How? It depends on what you have studied, which you haven't told us. You could do it with geometry, calculus with Jacobians, polar coordinates, vector functions?
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I presume that you mean that the region is all points in the plane such that (x, y)= (u cos(v), u sin(v)). Those are essentially polar coordinates!

Do you see that \(\displaystyle x^2+ y^2= u^2cos^2(v)+ u^2sin^2(v)= u^2\)? That is, that all (x, y) for a fixed u lie on the circle with center at the origin and radius u? Since u goes from 0 to 1/3, if v went from 0 to \(\displaystyle 2\pi\), that would be the disk, centered at the origin with radius 1/3. But since v goes from 0 to \(\displaystyle \pi/4\), this is a wedge taking up 1/8 of that disk. What is that area?
 
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