Area bounded by the curve $\ y = e^{x^2}$ and the lines $\ x = 1$ and $\ x = 2$ is equal to A sq.units. Find the area bounded by the curve $\ y = {\sqrt {\ln(x)}}$, $\ y$ axis and the lines $\ y = e$ and $\ y = e^4$ .
My attempt : I tried finding any relation between areas of both functions graphically as they are inverse of each other but i couldn't . Please help..
The area you are looking for is $2e^4 -e-\bf A$.
The graphic relationship between both areas is: \begin{align} \int_1^2 e^{x^2}\text{ d}x +\int_e^{e^4} \sqrt{\ln x}\text{ d}y &= (\text{area of a base-} 2 \text{, height-} e^{(2)^2} \text{ rectangle}) \\ &\qquad\text{ minus (area of a base-} 1 \text{, height-} e^{(1)^2} \text{ rectangle)} \\ &=2e^4-1e^1.\end{align}