Question

Solve the following :

Find the area of the circle x² + y² = 9, using integration.

Answer 1

By the symmetry of the circle, its area is equal to 4 times the area of the region OABO. Clearly for this region, the limits of integration are 0 and 3.

From the equation of the circle, y² = 9 – x².

In the first quadrant, y > 0

∴ y = `sqrt(9 - x^2)`

∴ Area of the circle = 4  ......(Area of the region OABO)

= `4int_0^3y.dx = 4int_0^3 sqrt(9 - x^2).dx`

= `4[x/2 sqrt(9 - x^2) + (9)/(2) sin^-1 (x/3)]_0^3`

= `4[3/2 sqrt(9 - 9) + (9)/(2) sin^-1 (3/3)] - 4[(0)/(2) sqrt(9 - 0) + (9)/(2)sin^1 (0)]`

= `4.(9)/(2).(pi)/(2)`

= 9π sq.units.