If the area bounded by the graph of `y=xe^(-ax)(agt0) and " the x-axis is "1//9` then find the value of a.

Answer

3

Explanation

`y=xe^(-ax)`
`therefore" "y'=e^(-ax)-xae^(-ax)`
`=e^(-ax)(1-ax)`
`"Let "y'=0`
`therefore" "x=(1)/(a)," which point of maxima".`
`underset(xrarroo)limxcdote^(-ax)=underset(xrarroo)lim(x)/(e^(a))=underset(xrarroo)lim(1)/(ae^(ax))=0`
`underset(xrarroo)limxcdote^(-ax)=-oo`

`"Area "=overset(oo)underset(0)intxe^(-ax)dx=[(xcdote^(-ax))/(-a)]_(0)^(oo)+(1)/(a)overset(oo)underset(0)inte^(-ax)dx`
`=(0)-[(1)/(a^(2))e^(-ax)]_(0)^(oo)`
`=-(1)/(a^(2))[0-1]=(1)/(a^(2))`
`"Given that "(1)/(a^(2))=(1)/(9)`
`rArr" "a=3`