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The area of the region bounded by the cu...

The area of the region bounded by the curve `y = 2x - x^2` and the line `y = x` is

A

`2overset(e)underset(1)intsqrt(log_(e)y)dy`

B

`2e-overset(1)underset(-1)inte^(x^(2))dx`

C

`overset(1)underset(-1)int (e-e^(x^(2)))dx`

D

`2overset(1)underset(0)intsqrt(x)e^(x)dx`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D

`"Area, "A=overset(1)underset(-1)int(e-e^(x^(2)))dx=2e-overset(1)underset(-1)inte^(x^(2))dx`
`y=e^(x^(2))`
`therefore" "log_(e)y=x^(2)`
`therefore" "x=pmsqrt(log_(e)y)`
`therefore" Area, "A=2overset(e)underset(1)intsqrt(log_(e)y)dy`
`"Now put "log_(e)y=t.`
`therefore" "y=e^(t)`
`therefore" "dy=e^(t)dt`
`"So, area "A=2overset(1)underset(0)intsqrt(t)" "e^(t)dt`
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