The line `x = (pi)/(4)` divides the area of the region bounded by y = sin x, y = cos x and X-axis `(0lexle(pi)/(2))` into two regions of areas `A_(1)` and `A_(2)`. Then, `A_(1) : A_(2)` equals

Options

A. `4 : 1`
B. `3 : 1`
C. `2 : 1`
D. `1 : 1`

Answer

D

Explanation

Area, `A_(1)=int_(0)^(pi//4)sinxdx`

`=-[cosx]_(0)^(pi//4)=1-(1)/(sqrt(2))=(sqrt(2)-1)/(sqrt(2))`
and area, `A_(2)=int_(pi//4)^(pi//2)cosxdx=[sinx]_(pi//4)^(pi//2)=[1-(1)/(sqrt(2))]=(sqrt(2)-1)/(sqrt(2))`
`therefore A_(1):A_(2)=(sqrt(2)-1)/(sqrt(2)):(sqrt(2)-1)/(sqrt(2))=1:1`