Question

Find the area of circle x² + y² = 25.

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 = 5.
Given equation of the circle is
x² + y² = 25
∴ y² = 25 – x²
∴ y = ± `sqrt(25 - x^2)`
∴ y = `sqrt(25 - x^2)`       ...[∵ In first quadrant , y > 0]
∴ Required area = 4 (area of the region OPQO)
= `4 xx int_0^5y*dx`

= `4 xx int_0^5 sqrt(25 - x^2)*dx`

= `4int_0^5 sqrt((5)^2 - x^2)*dx`

= `4[x/2 sqrt((5)^2 - x^2) + (5)^2/(2)sin^-1(x/5)]_0^5`

= `4{[5/2 sqrt(25 - (5)^2) + (25)/(2)sin^-1(5/5)] - [0/2 sqrt(25 - (0)^2) + (25)/(2)sin^-1(0/5)]}`

= `4{[5/2 (0) + 25 / 2sin^-1 (1)] - [0 + 0]}`

= `4(25 / 2 xx pi/(2))`
= 25π sq. units.