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Find the area enclosed between the curves |y| = 1 - x² and x² + y² = 1.

1. (3*pi - 8)/3 sq. units
2. (pi - 8)/3 sq. units
3. (2*pi - 8)/3 sq. units
4. None of these

Correct answer: (3*pi - 8)/3 sq. units

Solution

The curve |y| = 1-x² exists for |x|<=1 and gives y = 1-x² (for y>=0) and y = -(1-x²) (for y<=0). Intersections with unit circle x²+y²=1: in upper half, x²+(1-x²)²=1 => x⁴-x²=0 => x=0 or x=+-1. The enclosed area (between |y|=1-x² and circle) consists of two lens-shaped regions (top and bottom). Total area = area of unit circle - area enclosed by |y|=1-x² curve. Area under y=1-x² from -1 to 1 = integral[-1 to 1](1-x²)dx = 2 - 2/3 = 4/3 (top). By symmetry, bottom part also = 4/3. Total area inside |y| curve = 8/3. Area enclosed between curves = pi*1² - 8/3... but we need the region between the two curves. Enclosed area = 2*(semicircle area - area under parabola) = 2*(pi/2 - 4/3) = pi - 8/3. Wait: the problem says area enclosed BETWEEN the two curves, so the regions where one curve is inside the other. The circle encloses the parabola region, so the enclosed area between them = circle area - area inside |y|=1-x². Circle area = pi. Area inside |y|<=1-x² = 8/3. But area of circle = pi and 8/3 < pi (since pi~3.14 and 8/3~2.67). Area between = pi - 8/3... but none of the options equals pi-8/3 except we need to check option (c): (2*pi-8)/3 = 2*pi/3 - 8/3. Hmm. Let me redo: the region enclosed between the two curves consists of 4 "petal" like regions. Actually the parabola lies inside the circle. The area between the curves = (area of circle) - (area bounded by |y|=1-x²) = pi - 8/3 which doesn't match cleanly. The region common to both (inside both curves) would be |y|<=1-x² within |y|<=1, so the intersection is the parabolic region of area 8/3... Checking option (c): (2pi-8)/3. This equals 2*(pi-4)/3 = 2*(pi/2 - 2) * 2/3. Actually 2*(pi/2 - 4/3) = pi - 8/3 = (3pi-8)/3. So area = (3pi-8)/3... option (a). Verify: semicircle area = pi/2. Area under parabola (y=1-x² from -1 to 1) = [x - x³/3] from -1 to 1 = (1-1/3)-(-1+1/3) = 2/3 + 2/3 = 4/3. Area between upper semicircle and upper parabola = pi/2 - 4/3. By symmetry, double for both halves: 2*(pi/2 - 4/3) = pi - 8/3 = (3pi-8)/3. Answer: (3pi-8)/3.

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