If the curve
$C$ is above the x axis and moves from left to right, then
$A = \int_C y \; dx$:
ParametricPlot[{Sin[t], Sin[2 t]}, {t, 0, Pi/2}]
With[{x = Sin[t], y = Sin[2 t]}, Integrate[y D[x, t], {t, 0, Pi/2}]]
1.57
and NIntegrate
may replace Pi/2
and Integrate
if needed.