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Question
Find the area of the region bounded by y = `sqrt(x)` and y = x.
Solution
We are given the equations of curve y = `sqrt(x)` and line y = x.
Solving y = `sqrt(x)`
⇒ y2 = x and y = x,
We get x2 = x
⇒ x2 – x = 0
⇒ x(x – 1) = 0
∴ x = 0, 1
Required area of the shaded region
= `int_0^1 sqrt(x) "d"x - int_0^1 x "d"x`
= `2/3 [x^(3/2)]_0^1 - 1/2 [x^2]_0^1`
= `2/3[(1)^(3/2) - 0] - 1/2 [(1)^2 - 0]`
= `2/3 - 1/2`
⇒ `(4 - 3)/6`
⇒ `1/6` sq.units
Hence, the required area = `1/6` sq.units
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