Applications of Integration — Quiz 2

Questions in this quiz (10)

1. Find the volume of the solid whose base is a circle of radius $$4$$ centered at the origin and that has cross sections that are semicircles perpendicular to the x-axis.

   • $$8\pi$$
   • $$\frac{128\pi}{3}$$
   • $$12\pi$$
   • $$10\pi$$

2. Find the length of the curve $$y=\cos x$$ from $$x=0$$ to $$x=\pi$$.

   • $$2$$
   • $$\pi$$
   • $$2\sqrt2$$
   • $$3.8202$$

3. If a particle moves along the graph defined by $$x=\sin t$$ and $$y=\cos t$$, how far does the particle move from $$t=0$$ to $$t=\frac{\pi}{2}$$?

   • $$\frac{\pi}{2}$$
   • $$\frac{\pi}{4}$$
   • $$1$$
   • $$2$$

4. What is the length of the curve $$y=\ln(\cos x)$$ from $$x=0$$ to $$x=\frac{\pi}{4}$$?

   • $$\ln 2$$
   • $$\frac{\pi}{4}$$
   • $$\ln(\sqrt2+1)$$
   • $$\frac{\sqrt2}{2}$$

5. Find the arc length of $$y=\sqrt{x+2}$$ from $$x=-2$$ to $$x=2$$.

   • $$\frac{10}{3}$$
   • $$\sqrt{17}+\frac14\ln(4+\sqrt{17})$$
   • $$\frac{12}{3}$$
   • $$\frac{14}{3}$$

6. How long is the curve defined by $$x=y^3-2y$$ from $$y=-1$$ to $$y=1$$?

   • $$2$$
   • $$3.225828$$
   • $$\frac{10}{3}$$
   • $$3$$

7. Find the length of the parametric curve defined by $$x=t^2,\ y=t^3$$ from $$t=0$$ to $$t=2$$.

   • $$\frac{10}{3}$$
   • $$\frac{14}{3}$$
   • $$\frac{16}{3}$$
   • $$\frac{80\sqrt{10}-8}{27}$$

8. Find the perimeter of the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ using parametric equations.

   • $$15.86544$$
   • $$2\pi\sqrt{\frac{13}{2}}$$
   • $$6\pi$$
   • $$5\pi$$

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

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

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

    • $$\frac{\pi}{2}$$
    • $$\frac{3\pi}{8}+\frac12\tan^{-1}\left(\frac12\right)-\frac25$$
    • $$\frac{\pi}{4}$$
    • $$\frac{5\pi}{4}$$