The area of the region bounded between the curves y=ex|ln|...

The area of the region bounded between the curves $y=e\|x|\ln | x\|, x^{2}+y^{2}-2(|x|+|y|)+1 \geq 0$ and $X$ -axis where $|x| \leq 1$, if $\alpha$ is the $x$ -coordinate of the point of intersection of curves in 1st quadrant, is (a) $4\left[\int_{0}^{\alpha} e x \ln x d x+\int_{a}^{1}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$ (b) $4\left[\int_{0}^{a} e x \ln x d x+\int_{1}^{a}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$ (c) $4\left[-\int_{0}^{a} e x \ln x d x+\int_{\alpha}^{1}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$ (d) $2\left[\int_{0}^{a} e x \ln x d x+\int_{1}^{a}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$

The area of the region bounded between the curves $y=e\|x|\ln | x\|, x^{2}+y^{2}-2(|x|+|y|)+1 \geq 0$ and $X$ -axis
where $|x| \leq 1$, if $\alpha$ is the $x$ -coordinate of the point of intersection of curves in 1st quadrant, is
(a) $4\left[\int_{0}^{\alpha} e x \ln x d x+\int_{a}^{1}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$
(b) $4\left[\int_{0}^{a} e x \ln x d x+\int_{1}^{a}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$
(c) $4\left[-\int_{0}^{a} e x \ln x d x+\int_{\alpha}^{1}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$
(d) $2\left[\int_{0}^{a} e x \ln x d x+\int_{1}^{a}\left(1-\sqrt{1-(x-1)^{2}}\right) d x\right]$

Key Concepts

Area Between Curves

This concept involves computing the area of a region bounded by two or more curves. The idea is to set up definite integrals based on the vertical (or horizontal) distance between the curves over a specified interval. In problems like these, one must first identify the upper and lower boundaries of the region, then integrate their difference over the interval defined by the intersection points of the curves.

Curve Intersection

Determining the points where the bounding curves intersect is key to setting the limits of integration. In many problems, including this one, the intersection point in the first quadrant (or relevant section) provides a crucial boundary for dividing the region into simpler parts that can be integrated. Solving for the intersection involves equating the functions and considering the geometric restrictions (such as the first quadrant) applied by absolute value operations.

Symmetry in Integration

When a region exhibits symmetry, such as reflection across axes or about the origin, it can simplify the computation of area. By computing the area in one symmetric portion (e.g., the first quadrant) and then multiplying by an appropriate factor (like 4 for four symmetric regions), the overall calculation becomes easier. Recognizing symmetric properties helps in both setting up integrals and reducing computational complexity.

Handling Absolute Value Functions

Absolute value functions can split definitions into different cases over various intervals. Properly addressing these cases is important when setting up integrals, especially since the behavior of the function can change depending on whether the input is positive or negative. In the given problem, the use of |x| and |y| in defining the curves and regions necessitates careful consideration of piecewise behavior when determining bounds and expressions for integration.

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