When a curve is given parametrically as x = x(t), y = y(t), the area under the curve from t = t1 to t = t2 is A = |integral from t1 to t2 of y(t) * x'(t) dt|. For a closed curve traversed counterclockwise, A = $\frac{1}{2}$|integral of (xdy - ydx)|. Key application: the ellipse x = acos(t), y = bsin(t), t from 0 to 2pi. A = |integral from 0 to 2pi of bsin(t) * (-asin(t)) dt| = ab * integral from 0 to 2*pi of $sin^2$(t) dt = ab * pi. The sign convention requires care: if the parameter traces the curve clockwise, the integral gives a negative value, so always take the absolute value for area.