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Let lambda be the area of the region bounded by y >= cot(cot^(-1)(|ln(e^|x|)|)) and x² + y² - 6|x| - 6y + 9 <= 0. Find the value of lambda / pi.

1. 9/4 - 9*sqrt(2)/4
2. (9*pi - 18) / (4*pi)
3. 9/4
4. (9*pi/4 - 9/2) / pi

Correct answer: (9*pi/4 - 9/2) / pi

Solution

Step 1: cot(cot^(-1)(|ln(e^|x|)|)) = |x| (since cot(cot^(-1)(t)) = t for t >= 0 and |ln(e^|x|)| = |x|). So the region requires y >= |x|. Step 2: x² + y² - 6|x| - 6y + 9 <= 0. For x >= 0: (x-3)² + (y-3)² <= 9, circle centre (3,3) radius 3. For x < 0: (x+3)² + (y-3)² <= 9, circle centre (-3,3) radius 3. Step 3: In x >= 0 half: find the area of the part of circle centred (3,3) r=3 where y >= x. The line y = x passes through (3,3) (the centre), so it cuts the circle into two equal halves. Required area = half the circle = (1/2)*pi*9 = 9*pi/2. But the circle centre is at (3,3) and y = x passes through (3,3), so exactly half the circle lies above y = x. By symmetry the x < 0 half mirrors with y >= -x giving another 9*pi/2. However the two semicircles together give total area 9*pi/2. Subtracting the overlap at x=0 (none since circles touch at (0,3) which is on y >= |x| boundary). Total lambda = 9*pi/2.

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