1.

Consider the function defined implicity by the equation `y^(2)-2ye^(sin^(-1)x)+x^(2)-1+[x]+e^(2sin ^(-1)x)=0("where [x] denotes the greatest integer function").` The area of the region bounded by the curve and the line `x=-1` isA. `pi+1` sq. unitsB. `pi-1` sq. unitsC. `(pi)/(2)+1` sq. unitsD. `(pi)/(2)-1` sq. units

Answer» Correct Answer - A
`"For "-1lexlt0`
`(y-e^(sin^(-1)x))^(2)=2-x^(2)`
`y=e^(sin-1x)pmsqrt(2-x^(2))`
`A=overset(0)underset(-1)int(e^(sin^(-1)x)+sqrt(2-x^(2)))-(e^(sin^(-1)x)-sqrt(2-x^(2)))dx`
`=2overset(0)underset(-1)intsqrt(2-x^(2))dx`
`=2((1)/(2)xsqrt(2-x^(2)) :|_(-1)^(0)+(2)/(2)sin^(-1)""(x)/(sqrt(2)):|_(-1)^(0))`
`=[1+2(0-(-(pi)/(54)))]`
`=(pi)/(2)+1` sq. units.
`"For "0lexlt1,y=sin^(-1)xpmsqrt(1-x^(2))`
`A=2overset(1)underset(0)intsqrt(1-x^(2))dx`
`=2[(x)/(2)sqrt(1-x^(2)):|_(0)^(1)+(1)/(2)sin^(-1)""(x)/(1):|_(0)^(1)]`
`=0+sin^(-1)(1)=(pi)/(2)` sq. units.
`"Total area "=((pi)/(2)+1)+(pi)/(2)=pi+1.`


Discussion

No Comment Found