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If the line x=alpha divides the area of ...

If the line `x=alpha` divides the area of region `R={(x,y) in R^(2): x^(3)leylex, 0le xle 1}` into two equal parts, then

A

2

B

`(4)/(3)`

C

`(1)/(3)`

D

`(2)/(3)`

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`A={(x,y): 0 le y le x|x|+1 and -1 le x le 1}`
When `x ge 0," then " 0 le y le x^(2) +1`
and when `x lt 0," then " 0 le y le -x^(2)+1`
Now, the required region is the shaded region.

`[ because y=x^(2)+1 rArr x^(2)=(y-1),` parabola with vertex (0, 1) and
`y= -x^(2)+1 rArr x^(2)= -(y-1),`
Parabola with vertex (0, 1) but open downward ]
We need to calculate the shaded area, which is equal to
`int_(-1)^(0)(-x^(2)+1)dx+int_(0)^(1)(x^(2)+1)dx`
`=[-(x^(3))/(3)+x]_(-1)^(0)+[(x^(3))/(3)+x]_(0)^(1)`
`=(0-[-((-1)^(3))/(3)+(-1)])+([(1)/(3)+1]-0)`
`=-((1)/(3) -1)+(4)/(3)`
`=(2)/(3)+(4)/(3)=2`
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