Applications of Integration — Quiz 3

Questions in this quiz (9)

1. Show that the area of a quarter circle of radius $$3$$ equals $$\frac{9\pi}{4}$$ using polar area.

   • $$\frac{9\pi}{2}$$
   • $$\frac{9\pi}{4}$$
   • $$3\pi$$
   • $$\frac{3\pi}{2}$$

2. Find the area bounded by the graphs $$r=2\sin\theta$$ and $$r=1+\cos\theta$$.

   • $$\frac{\pi}{2}$$
   • $$\frac{3\pi}{4}$$
   • $$\pi$$
   • $$\frac{3\pi}{8}+\frac12\tan^{-1}\left(\frac12\right)-\frac25$$

3. Find the area bounded by $$r=-3\sin\theta$$.

   • $$\frac{9\pi}{4}$$
   • $$\frac{9\pi}{2}$$
   • $$3\pi$$
   • $$\frac{3\pi}{2}$$

4. Find the area in the second quadrant bounded by $$r=\sin\theta+\cos\theta$$.

   • $$\frac{\pi}{8}$$
   • $$\frac{\pi}{4}$$
   • $$\frac{\pi}{8}+\frac14$$
   • $$\frac{\pi}{2}$$

5. Find the area enclosed by the graph of $$r=\sin 2\theta$$.

   • $$\frac{\pi}{4}$$
   • $$\frac{\pi}{2}$$
   • $$\frac{3\pi}{4}$$
   • $$\pi$$

6. Find the area of the region bounded by $$r=1+\sin\theta$$ and $$r=1$$.

   • $$\frac{\pi}{2}$$
   • $$\frac{3\pi}{4}$$
   • $$\pi$$
   • $$2+\frac{\pi}{4}$$

7. Find the area bounded by $$r=2-2\sin\theta$$ and $$r=3\cos\theta$$.

   • $$1.913223$$
   • $$\frac{3\pi}{4}$$
   • $$\pi$$
   • $$\frac{5\pi}{4}$$

8. Find the perimeter of the region bounded by $$x=y^2-2y-3$$ and the y-axis.

   • $$4+\frac{8\sqrt2}{3}$$
   • $$6+\frac{8\sqrt3}{3}$$
   • $$4+2\sqrt2+\frac12\ln(3+2\sqrt2)$$
   • $$3+\frac{8\sqrt2}{3}$$

9. A machine part is made from an alloy that costs $$150$$ per cm$$^3$$. The base of the part is bounded by the area between $$y=2x$$ and $$y=x^2$$. The part can be made using either semicircular or square cross sections perpendicular to the x-axis. Which method is less expensive?

   • Semicircular cross sections
   • Semicircular cross sections
   • Same cost
   • Cannot be determined