Find the area of the region bounded by the x-axis and the curves defined by `y=tanx,-pi/3 le x le pi/3` and `y=cotx, pi/6 le x le (3pi)/2`

Answer

`((1)/(2) log_(e)3)` sq units

Explanation

Given, `y={(tanx",", -(pi)/(3) le x le (pi)/(3)),(cot x",",(pi)/(6) le x le (pi)/(2)):}`
which could be plotted as Y-axis.

` therefore "Required area " =int_(0)^(pi//4)(tanx)dx + int_(pi//4)^(pi//3) (cotx)dx`
`=[-log|cosx|]_(0)^(pi//4) + [log sin x]_(pi//4)^(pi//3)`
`= -("log" (1)/(sqrt(2)) -0)+("log"(sqrt(3))/(2)-"log"(1)/(sqrt(2)))`
`="log"(sqrt(3))/(2)-2"log"(1)/(sqrt(2))`
`="log"(sqrt(3))/(2)-"log"(1)/(2)=((1)/(2) log_(e)3)` sq units