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Find the area of the region bounded b...

Find the area of the region bounded by the parabola `y=x^2` and `y" "=" "|x|` .

Text Solution

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given curve is `y=x^2`...(i)
`y=∣x∣`..... (ii)
Curve (i) is symmetrical about y-axis since it contains even power of x. Solving (1) and (2), we get
Case 1: If `x>=0,x^2=x⇒x(x−1)=0`
Therefore, `x=0,x=1`
So, the point intersections are`(0, 0)` and `(1,1)`
Case 2: If `x<=0,x^2=−x⇒x(x+1)=0` Therefore `x=0,x=−1` So, point of intersections are `(0,0)` and `(−1,1).`
Now required area = `2xx`ar(OABCO)
= `2[ int_0^1|x| dx - int_0^1 x^2 dx]`
= `2[int_0^1 ( x- x^2) dx]`
= `2[ x^2/(2) - x^3/(3)]_0^1`

= `1/3` units
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