Answer
$\frac{e^{3\pi}-\sqrt{e^{3\pi}}}{4}$
Work Step by Step
The area of the region is determined by the integral $A=\int_{3\pi/4}^{3\pi/2}\frac{1}{2}r^2d\theta$,
Evaluate $A$:
$A=\int_{3\pi/4}^{3\pi/2}\frac{1}{2}(e^{\theta})^2d\theta$
$A=\int_{3\pi/4}^{3\pi/2}\frac{1}{2}e^{2\theta}d\theta$
$A=[\frac{e^{2\theta}}{4}]_{3\pi/4}^{3\pi/2}$
$A=\frac{e^{3\pi/2\cdot 2}}{4}-\frac{e^{3\pi/4\cdot 2}}{4}$
$A=\frac{e^{3\pi}}{4}-\frac{e^{3\pi/2}}{4}$
$A=\frac{e^{3\pi}-\sqrt{e^{3\pi}}}{4}$
Thus, the area is $\frac{e^{3\pi}-\sqrt{e^{3\pi}}}{4}$.