Area under curve and its inverse

Question

Area bounded by the curve $y=e^{x^2}$, $x$ axis and the lines $x=1$, $x=2$ is given to be equal to $a$ (unit$^2$). Area bounded by the curve $y=\sqrt{\ln x}$, $y$ axis, and the lines $y=e$ and $y=e^4$ is equal to what? (in terms of $a$)

Answer 1

The functions corresponding to two curves are inverse functions. To see this, let $f(x)=e^{x^2}$ and $g(x)=\sqrt{\ln x}$ $$f(g(x))=e^{\ln x}=x \quad \Rightarrow\quad g(x)=f^{-1}(x)$$ Hence $$\int_{e}^{e^4}g^{-1}(y)dy=\int_{f^{-1}(e)}^{f^{-1}(e^4)}f(x)dx=\int_{1}^{2}f(x)dx=a$$

Viola! The two areas are actually equal. :)

(Note that we don't need absolute value to compute area because functions are nonnegative on relevant intervals)