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Question
The area bounded by the curve y = x | x|, x-axis and the ordinates x = –1 and x = 1 is given by ______.
[Hint: y = x2 if x > 0 and y = –x2 if x < 0]
Options
0
`1/3`
`2/3`
`4/3`
Solution
The area bounded by the curve y = x | x|, x-axis and the ordinates x = –1 and x = 1 is given by `underline(2/3)`.
Explanation:
When x > 0, |x| = x
∴ Equation of the curve y = x2
When x < 0, |x| = -x
Equation of the curve y = -x2
Curve y = x |x|, x ≥ -1, x ≤ 0
The area bounded by x-axis = Area of region APO + Area of region OBQ
`= |int_(-1)^0 - x^2 dx| + int_0^1 x^2 dx`
`= |[-x^3/3]|_-1^0 + [x^3/3]_0^1`
`= |-0 - 1/3| + [1/3 - 0]`
`= 1/3 + 1/3`
`= 2/3` square unit
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