Without a picture I can only guess what the problem is.
Let’s assume that the circle is inscribed in a square touching each of its sides.
The side of the square is the square root of its area=8x⁶+1. Let’s assume also that the area of the circle is 49πy². From this the diameter of the circle is 14y and 14y=8x⁶+1, so y=(8x⁶+1)/14.
If the shaded area is the area between the circle and the square the area of the shaded region is:
(8x⁶+1)²-49πy². But y²=(8x⁶+1)²/196, so the shaded area is (8x⁶+1)²-49π(8x⁶+1)²/196.
This simplifies to (8x⁶+1)²(1-π/4) m².
But there’s a coincidence. 8x⁶+1=(2x²+1)(4x⁴-2x²+1), and this may be a clue to something which a picture might resolve. It may show a different relationship between x and y.
If the area of the circle is 49y², rather than 49πy², then, if r=radius, πr²=49y² and r²=49y²/π, r=7y/√π.
So 14y/√π=8x⁶+1, y=(8x⁶+1)√π/14 and y²=(8x⁶+1)²π/196.
The shaded area is (8x⁶+1)²-49y²=(8x⁶+1)²(1-π/4) m², same as before. The reason is that a given square can only have one inscribed circle. This leads me to believe that the picture would not show an inscribed circle, but would relate the square and circle in another way. The “coincidence” identified earlier I suspect is a major clue to a special relationship between x and y, which a picture could well resolve.