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Question
Find the area of a minor segment of the circle x2 + y2 = a2 cut off by the line x = `"a"/2`
Solution
Solving the equation x2 + y2 = a2 and x = `"a"/2`
We obtain their points of intersection which are `("a"/2, sqrt(3) "a"/2)` and `("a"/2, - (sqrt(3)"a")/2)`.
Hence, From the figure in the question, we get
Required Area = 2 Area of OAB
= `2 int_("a"/2)^"a" sqrt("a"^2 - x^2) "d"x`
= `2[x/2 sqrt("a"^2 - x^2) + "a"^2/2 sin^-1 x/"a"]_("a"/2)^"a"`
= `2["a"^2/2 * pi/2 - "a"/4 * "a" sqrt(3)/2 - "a"^2/2 * pi/6]`
= `"a"^2/12 (6pi - 3sqrt(3) - 2pi)`
= `"a"^2/12 (4pi - 3sqrt(3))` sq.units
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