If ∫₀² ( √2x - √(2x - x²) )dx = ∫₀¹ ( 1 - √(1 - y²) - y²/2 )dy + ∫₁² ( 2 - y²/2 )dy + I , then I equals

Solution

Maths sir · 5 steps

Read the left side as an area, then slice it horizontally: each strip is right curve minus left curve.

1. Name the curves. y=√2x is the parabola x=y²/2. y=√(2x-x²) is the upper half of the circle (x-1)²+y²=1, that is x=1±√(1-y²). The left side is the area between them for 0≤ x≤ 2, with the parabola on top.
2. Lower strips. For 0≤ y≤ 1, a horizontal line at height y cuts the region into two pieces. The left piece runs from the parabola to the left arc and has width 1-√(1-y²)-y²/2, which is the first given integral. The right piece runs from the right arc x=1+√(1-y²) to the line x=2 and has width 1-√(1-y²).
3. Upper strips. For 1≤ y≤ 2, each strip runs from the parabola to x=2, so its width is 2-y²/2. That is the second given integral, and it ends at y=2, where the parabola meets x=2. The only piece left over is the right-hand strip from y=0 to y=1.

   I=∫₀¹(1-√(1-y²))dy

4. Check the numbers. The left side is 8/3-π/2: the parabola part gives 8/3 and the semicircle gives π/2. The two given integrals are 5/6-π/4 and 5/6. So I=8/3-π/2-5/3+π/4=1-π/4, which is exactly ∫₀¹(1-√(1-y²))dy.
5. So. The answer is (c), I=∫₀¹(1-√(1-y²))dy.

The idea to keep. When an area is rewritten in terms of y, draw the region and slice it horizontally. Each strip is right curve minus left curve, and a strip that splits into two pieces needs two integrals.

Answer from the NTA final answer key. Solution by PrepWiser.