The area bounded by the curves y= x, y=|x|, y=| x| and y=||x|| is:

Question

The area bounded by the curves $y=\ln x$, $y=\ln|x|$, $y=|\ln x|$ and $y=|\ln|x||$ is:

Accepted Answer

The graph of $y=|f(x)|$ is obtained by reflecting, above the $x$-axis, the portion of $y=f(x)$ that lies below it. Combining the four curves and using symmetry about both axes:

$$\text{Area} = 4\int_0^1(-\ln x)\,dx = -4\left[x\ln x - x\right]_0^1 = -4(0-1) = 4 \text{ sq. units}$$

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