AP Polar and Parametric Review for BC Calc

What is the slope of the line tangent to the polar curve $r=1+2\sin\theta$ at  $\theta=0$ ?

• 2

• 1/2

• 0

• - 1/2

• - 2

Calculator:  The figure shows the graphs of the polar curves $r=2\cos\left(3\theta\right)$ and  $r=2$ .  What is the sum of the areas of the shaded regions? 

• 0.858

• 3.142

• 8.566

• 9.425

• 15.708

The function $f\left(\theta\right)=2\theta^3-20\theta^2+50\theta$ satisfies  $f\left(\theta\right)\ge0$ for $\theta\ge0$ . During the time interval  $0\le t\le2\pi$ seconds, a particle moves along the polar curve  $r=f\left(\theta\right)$ so that at time t seconds,  $\theta=t$ .  On what interval(s) of time is the distance between the particle and the pole increasing?    

•  $\left[0,2\pi\right]$  

•  $\left[0,5\right]$  

•  $\left[\frac{5}{3},5\right]$  

•  $\left[0,\frac{5}{3}\right],\ \left[5,2\pi\right]$  

Which of the following is equal to the area of the region inside the polar curve $r=2\cos\theta$ and outside the polar curve $r=\cos\theta$ ? 

•  $3\int_0^{\frac{\pi}{2}}\left(\cos^2\theta\right)d\theta$  

•  $3\int_0^{\pi}\left(\cos^2\theta\right)d\theta$  

•  $\frac{3}{2}\int_0^{\frac{\pi}{2}}\left(\cos^2\theta\right)d\theta$  

•  $3\int_0^{\frac{\pi}{2}}\left(\cos\theta\right)d\theta$  

•  $3\int_0^{\pi}\left(\cos\theta\right)d\theta$  

The area of the region inside the polar curve $r=4\sin\theta$ and outside the polar curve $r=2$ is given by... 

•  $\frac{1}{2}\int_0^{\pi}\left(4\sin\theta-2\right)^2d\theta$  

•  $\frac{1}{2}\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}\left(4\sin\theta-2\right)^2d\theta$  

•  $\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}\left(4\sin\theta-2\right)^2d\theta$  

•  $\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}}\left(16\sin^2\theta-4\right)d\theta$  

•  $\frac{1}{2}\int_0^{\pi}\left(16\sin^2\theta-4\right)d\theta$  

Which of the following expressions gives the total area enclosed by the polar curve $r=\sin^2\theta$ shown in the figure? 

•  $\frac{1}{2}\int_0^{\pi}\left(\sin^2\theta\right)d\theta$  

•  $\int_0^{\pi}\left(\sin^2\theta\right)d\theta$  

•  $\frac{1}{2}\int_0^{\pi}\left(\sin^4\theta\right)d\theta$  

•  $\int_0^{\pi}\left(\sin^4\theta\right)d\theta$  

•  $2\int_0^{\pi}\left(\sin^4\theta\right)d\theta$  

What is the area of the region enclosed by the lemniscate $r^2=18\cos\left(2\theta\right)$ shown in the figure? 

• 9/2

• 9

• 18

• 24

• 36

Which of the following gives the area of the region enclosed by the loop of the graph of the polar curve  $r=4\cos\left(3\theta\right)$ as shown in the figure? 

•  $16\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\cos\left(3\theta\right)d\theta$  

•  $8\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\cos\left(3\theta\right)d\theta$  

•  $\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}}\cos^2\left(3\theta\right)d\theta$  

•  $16\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\cos^2\left(3\theta\right)d\theta$  

•  $8\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}}\cos^2\left(3\theta\right)d\theta$  

Which of the following represents the graph of the polar curve $r=2\sec\theta$  

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Calculator:  The figure shows the graph of $r=0.3\theta^2-\sin\left(2\theta\right)$ on the interval $\left[0,\frac{\pi}{2}\right]$ .  Let S be the region bounded by the enclosed loop in the figure.

Find the area of S of the enclosed loop. 

• 1.103

• 0.235

• 0.514

• 0.985