, 2024

To calculate the shaded area, we add the area of the rectangle
and the areas of the four semi-circles, and subtract the area of the
larger circle.

Since the rectangle is $6$ by $8$, its area is $6 \times 8 = 48$.

The two semi-circles of diameter $6$
together form a complete circle of diameter $6$, or radius $3$. The combined area of these
semi-circles is $\pi \times 3^2$ or
$9\pi$.

The two semi-circles of diameter $8$
together form a complete circle of diameter $8$, or radius $4$. The combined area of these
semi-circles is $\pi \times 4^2$ or
$16\pi$.

Since the larger circle passes through the four vertices of the
rectangle, the diagonal of the rectangle is its diameter. (This is
because the diagonal subtends an angle of $90\degree$ at each of the other vertices
and so is a diameter.)

The length of the diagonal is $\$\sqrt{6^2}$ +
8^2} = $\sqrt{100}$ = 10$, and so the radius of the larger circle
is $5$, and so its area is $\pi \times 5^2 = 25\pi$.

Finally, this means that the area of the shaded region is $48 + 9 \pi + 16 \pi - 25 \pi$ which equals
$48$.

The closest integer to $48$ is $48$.