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Question
Find the area bounded by the curve y = `sqrt(x)`, x = 2y + 3 in the first quadrant and x-axis.
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Solution
Given that: y = `sqrt(x)`, x = 2y + 3, first quadrant and x-axis.
Solving y = `sqrt(x)` and x = 2y + 3
We get y = `sqrt(2y + 3)`
⇒ y2 = 2y + 3
⇒ y2 – 2y – 3 = 0
⇒ y2 – 3y + y – 3 = 0
⇒ y(y – 3) + 1(y – 3) = 0
⇒ (y + 1)(y – 3) = 0
∴ y = –1, 3
Area of shaded region
= `int_0^3 (2y + 3) "d"y - int_0^3 "y"^2 "d"y`
= `[2 y^2/2 + 3y]_0^3 - 1/3 [y^3]_0^3`
= `[(9 + 9) - (0 + 0)] - 1/3[27 - 0]`
= 18 – 9
= 9 sq.units
Hence, the required area = 9 sq.units
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