Let `S` be the area bounded by the curve `y=sinx(0lt=xlt=pi)` and the x-axis and `T` be the area bounded by the curves `y=sinx(0lt=xlt=pi/2),y=acosx(0lt=xlt=pi/2),` and the x-axis `(w h e r ea in R^+)` The value of `(3a)` such that `S : T=1:1/3` is______

Answer

4

Explanation

We have `S=overset(pi)underset(0)sin x dx =2, so T=(2)/(3)," where "agt0.`
`"Now "T=overset(tan^(-1)a)underset(0)intsin x dx +overset(pi//2)underset(tan^(-1)a)inta cos x dx =(2)/(3)`

`i.e., -cos (tan^(-1)a)+1+a{[1-sin (tan^(-1)a)]}=(2)/(3)`
`i.e., -(1)/(sqrt(1+x^(2)))+1+a-(a^(2))/(sqrt(1+a^(2)))=(2)/(3)`
`"or "(a+1)-sqrt(a^(2)+1)=(2)/(3)or a +(1)/(3)=sqrt(a^(2)+1)`
`"or "a=(4)/(3)`
`"Hence, "3a=4`.