Question

Find the area of the region above the x-axis, included between the parabola y² = ax and the circle x² + y² = 2ax.

Answer 1

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