4. The maximum area of an inscribed trapezoid with the major axis of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ as its base is ( ).
(A) $\frac{3 \sqrt{3}}{4} a b$
(B) $\frac{\sqrt{3}}{2} a b$
(C) $\frac{\sqrt{3}}{6} a^{2}$
(D) $\frac{\sqrt{3}}{8} a^{2}$
4. (A).

Let the base be $P P^{\prime}$, and $P(a \cos \theta, b \sin \theta)$, $P^{\prime}(-a \cos \theta, b \sin \theta)$.
$$
\begin{array}{l}
S_{\text {controlled }}=\frac{2 a+2 a \cos \theta}{2} b \sin \theta=a b(1+\cos \theta) \sin \theta \\
=a b \sqrt{(1+\cos \theta)^{2}\left(1-\cos ^{2} \theta\right)} \\
=\frac{a b}{\sqrt{3}} \sqrt{(1+\cos \theta)^{3}(3-3 \cos \theta)} \\
\leqslant \frac{a b}{\sqrt{3}}\left(\frac{1+\cos \theta+1+\cos \theta+1+\cos \theta+3-3 \cos \theta}{4}\right)^{2} \\
=\frac{a b}{\sqrt{3}} \cdot\left(\frac{3}{2}\right)^{2}=\frac{3 \sqrt{3} a b}{4} .
\end{array}
$$
(When $1+\cos \theta=3-3 \cos \theta$, i.e., $\cos \theta=\frac{1}{2}$, $S_{\text {group }}$ )