Question

Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32 is ______.

Options

• 16π sq.units

• 4π sq.units

• 32π sq.units

• 24 sq.units

Answer 1

Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32 is 4π sq.units.

Explanation:

Given equation of circle is x² + y² = 32

⇒ x² + y² = `(4sqrt(2))^2` and the line is y = x and the x-axis.

Solving the two equations 

We have x² + x² = 32

⇒ 2x² = 32

⇒ x² = 16

∴ x = ± 4

Required area = `int_0^4 x  "d"x + int_4^(4sqrt(2)) sqrt((4sqrt(2))^2 - x^2)  "d"x`

= `1/2 [x^2]_0^4 + [x/2 sqrt((4sqrt(2))^2 - x^2) + 32/2 sin^-1  x/(4sqrt(2))]_4^(4/sqrt(2))`

= `1/2 [16 - 0] + [0 + 16 sin^-1 ((4sqrt(2))/(4sqrt(2))) - 2sqrt(32 - 16) - 16sin^-1  4/(4sqrt(2))]`

= `8 + [16 sin^-1 (1) - 8 - 16sin^-1  1/sqrt(2)]`

= `8 + 16 * pi/2 - 8 - 16 * pi/4`

= `8pi - 4pi`

= 4π sq.units