find the area of the shaded region.\nr = 4 + 3 sin(θ)

Explanation:

Step1: Recall the formula for the area of a polar curve

The area ( A ) of a polar curve ( r = f(\theta) ) is given by ( A=\frac{1}{2}\int_{\alpha}^{\beta}r^{2}d\theta ). Here, the shaded region is from ( \theta=-\frac{\pi}{2} ) to ( \theta=\frac{\pi}{2} ) (since it is the right - hand side of the curve ( r = 4 + 3\sin\theta )), and ( r = 4+3\sin\theta ). So, ( A=\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(4 + 3\sin\theta)^{2}d\theta ).

Step2: Expand the integrand

Expand ( (4 + 3\sin\theta)^{2}=16+24\sin\theta + 9\sin^{2}\theta ). Then the integral becomes ( A=\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(16+24\sin\theta + 9\sin^{2}\theta)d\theta ).

Step3: Use the property of definite integrals for odd and even functions

• Recall that ( y = \sin\theta ) is an odd function (( \sin(-\theta)=-\sin\theta )), so ( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}24\sin\theta d\theta=0 ).
• Recall the double - angle formula ( \sin^{2}\theta=\frac{1 - \cos(2\theta)}{2} ). Then ( 9\sin^{2}\theta=\frac{9(1 - \cos(2\theta))}{2}=\frac{9}{2}-\frac{9\cos(2\theta)}{2} ).
• The integral now is ( A=\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}16d\theta+\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{9}{2}d\theta-\frac{1}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{9\cos(2\theta)}{2}d\theta ).
• ( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}16d\theta=16\left[\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}=16\pi ).
• ( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{9}{2}d\theta=\frac{9}{2}\left[\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}=\frac{9\pi}{2} ).
• For ( \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{9\cos(2\theta)}{2}d\theta ), let ( u = 2\theta ), ( du=2d\theta ). When ( \theta=-\frac{\pi}{2},u =-\pi ); when ( \theta=\frac{\pi}{2},u=\pi ). And ( y=\cos u ) is an even function (( \cos(-u)=\cos u )), but ( \int_{-\pi}^{\pi}\frac{9\cos u}{4}du=\frac{9}{4}[\sin u]_{-\pi}^{\pi}=0 ).

Step4: Calculate the area

( A=\frac{1}{2}(16\pi+\frac{9\pi}{2})=\frac{32\pi + 9\pi}{4}=\frac{41\pi}{4} ).

Answer:

(\frac{41\pi}{4})