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Question
Find the area of the region above the x-axis, included between the parabola y2 = ax and the circle x2 + y2 = 2ax.
Solution
Solving the given equations of curves
We have x2 + ax = 2ax or x = 0, x = a
Which give y = 0.
y = ±a
From the figure in the question
Area ODAB = `int_0^"a" (sqrt(2"a"x - x^2) - sqrt("a"x))"d"x`
Let x = 2a sin2θ.
Then dx = 4a sinθ cosθ dθ and x = 0
⇒ θ = 0, x = a
⇒ θ = `pi/4`.
Again, `int_0"a" sqrt(2"a"x - x^2) "d"x = int_0^(pi/4) (2"a" sintheta costheta)(4"a" sintheta costheta)"d"theta`
= `"a"^2 int_0^(pi/4) (1 - cos 4theta) "d"theta`
= `"a"^2(theta - (sin 4theta)/4)_0^(pi/4)`
= `pi/4 "a"^2`.
Further more,
`int_0^"a" sqrt("a"x) "d"x = sqrt("a") 2/3 (x^(3/2))_0^"a"`
= `2/3 "a"^2`
Thus the required area = `pi/4 "a"^2 - 2/3 "a"^2`
= `"a"^2 (pi/4 - 2/3)` sq.units
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