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Find the area bounded by the curve x^(2)...

Find the area bounded by the curve `x^(2) = 4y` and the line `x = 4y -2`.

A

`(7)/(8)`

B

`(9)/(8)`

C

`(5)/(4)`

D

`(3)/(4)`

Text Solution

Verified by Experts

The correct Answer is:
B
Given equation of curve is `x^(2)=4y,` which represent a parabola with vertex (0, 0) and it open upward.

Now, let us find the points of intersection of `x^(2)=4y` and `4y=x+2`.
For this consider, `x^(2) =x+2`
`rArr x^(2)-x-2=0`
`rArr (x-2)(x+1)=0`
`rArr x= -1," then " y=(1)/(4)`
and when `x=2, " then " y=1`
Thus, the points of intersection are `A(-1,(1)/(4)) and B(2,1).`
Now, required area = area of shaded region
`=int_(-1)^(2){y("line")-y("parabola")}dx`
`=int_(-1)^(2)((x+2)/(4)-(x^(2))/(4))dx=(1)/(4)[(x^(2))/(2)+2x-(x^(3))/(3)]_(-1)^(2)`
`=(1)/(4)[(2+4-(8)/(3))-((1)/(2)-2+(1)/(3))]`
`=(1)/(4)[8-(1)/(2)-3]=(1)/(4)[5-(1)/(2)]=(9)/(8)` sq units.
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