Question

Determine the area under the curve y = `sqrt("a"^2 - x^2)` included between the lines x = 0 and x = a.

Answer 1

Here, we are given y = `sqrt("a"^2 - x^2)`

⇒ y² = a² – x²

⇒ x² + y² = a²

Area of the shaded region

= `2[(1)^(3/2) - 0] - 3/2[(1)^2 - 0]`

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

= `["a"/2 sqrt("a"^2 - "a"^2) + "a"^2/2 sin^-1  "a"/"a" - 0 - 0]`

= `"a"^2/2 sin^-1 (1)`

= `"a"^2/2 * pi/2`

= `(pi"a"^2)/4`

Hence, the required area = `(pi"a"^2)/4` sq.units.