What is Fourier transform in terms of area under the curve? We know that integration gives area under the curve.  And that FT is also an integration.  How do we interpret FT in terms of the area under the curve especially because e^(jwt) is a complex term?  In general, while dealing with complex terms in integration can we relate to some area?
Since

$$X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt$$

you get for the real part of $X(\omega)$ (assuming that $x(t)$ is real)

$$X_R(\omega)=\int_{-\infty}^{\infty}x(t)\cos(\omega t)dt\tag{1}$$

and for the imaginary part

$$X_I(\omega)=-\int_{-\infty}^{\infty}x(t)\sin(\omega t)dt\tag{2}$$

So if you like, from (1) and (2) you can interpret the real part of the Fourier transform as the area under the curve $x(t)\cos(\omega t)$, and the imaginary part would then be the area under the curve $-x(t)\sin(\omega t)$.

Note that interpreting an integral as the area under a curve does not always help intuition. In the case of the Fourier transform it is more natural to interpret the integrals (1) and (2) as inner products.