Question

Find the area of the circle x² + y² = 16

Answer 1

By the symmetry of the circle, required area of the circle is 4 times the area of the region OPQO.

For the region OPQO, the limits of integration are x = 0 and x = 4.

 Given equation of the circle is x² + y² = 16

∴ y² = 16 – x²

∴ y = `+- sqrt(16 - x^2)`

∴ y = `sqrt(16 - x^2)`   ......[∵ In first quadrant, y > 0]

∴ Required area = 4(area of the region OPQO)

= `4 xx int_0^4 y*"d"x`

= `4 xx int_0^4 sqrt(16 - x^2)  "d"x`

= `4int_0^4 sqrt((4)^2 - x^2)  "d"x`

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

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

= `4{[0 + 8 sin^-1 (1)] - [0 + 0]}`

= `4(8 xx pi/2)`

= 16π sq.units