The area of the region, inside the ellipse $$x^{2}+4y^{2}=4$$ and outside the region bounded by the curves y=|x|-1 and y=1-|x|, is:

1. Area of the Ellipse

First, let's rewrite the equation of the ellipse in standard form by dividing by 4:

$$\frac{x^2}{4} + \frac{4y^2}{4} = \frac{4}{4} \implies \frac{x^2}{2^2} + \frac{y^2}{1^2} = 1$$

• The semi-major axis is $$a = 2$$.
• The semi-minor axis is $$b = 1$$.
• The area of an ellipse is given by the formula $$A = \pi ab$$.
• $$y = |x| - 1$$ (A "V" shape opening upwards with vertex at $$(0, -1)$$)
• $$y = 1 - |x|$$ (An inverted "V" shape with vertex at $$(0, 1)$$)
• $$(0, 1)$$ and $$(0, -1)$$ — Length of vertical diagonal $$(d_1) = 2$$
• $$(1, 0)$$ and $$(-1, 0)$$ — Length of horizontal diagonal $$(d_2) = 2$$

Total Ellipse Area $$= \pi(2)(1) = \mathbf{2\pi}$$

2. Area of the Inner Region

The inner region is bounded by two absolute value curves:

When you plot these, they intersect at $$(\pm 1, 0)$$. Together, they form a rhombus (specifically a square rotated $$45^\circ$$) with vertices at:

The area of a rhombus is $$\frac{1}{2} \times d_1 \times d_2$$:

Inner Region Area $$= \frac{1}{2} \times 2 \times 2 = \mathbf{2}$$

The question asks for the area inside the ellipse but outside the bounded region. This is simply the difference between the two areas:

$$\text{Required Area} = \text{Area of Ellipse} - \text{Area of Rhombus}$$

$$\text{Required Area} = 2\pi - 2$$

$$\text{Required Area} = \mathbf{2(\pi - 1)}$$