write an integral that represents the area of the shaded region of the figure. do not evaluate the…

Explanation:

Step1: Recall area formula in polar coordinates

The area $A$ of a polar - curve $r = f(\theta)$ is given by $A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta$.

Step2: Determine the limits of integration

For the polar curve $r = 3\cos(2\theta)$, we want to find the limits of integration for the shaded region. The curve $r = 3\cos(2\theta)$ is a four - leaf rose. To find the limits for the left - hand leaf (the shaded region), we set $r = 0$. So, $3\cos(2\theta)=0$, which gives $2\theta=\pm\frac{\pi}{2}+2k\pi$, or $\theta=\pm\frac{\pi}{4}+k\pi$. For the left - hand leaf, the lower limit $\alpha=\frac{3\pi}{4}$ and the upper limit $\beta=\frac{5\pi}{4}$.

Step3: Substitute $r$ into the area formula

Since $r = 3\cos(2\theta)$, then $r^{2}=9\cos^{2}(2\theta)$. The area integral is $A=\frac{1}{2}\int_{\frac{3\pi}{4}}^{\frac{5\pi}{4}}(9\cos^{2}(2\theta))d\theta$.

Answer:

$\frac{1}{2}\int_{\frac{3\pi}{4}}^{\frac{5\pi}{4}}(9\cos^{2}(2\theta))d\theta$