Applications of Integration — Quiz 1

Questions in this quiz (10)

1. Calculate the area bounded by $$f(x)=x^4-4x^2$$ and the x-axis. What important conclusion can be drawn from this problem?

   • Area equals total signed integral
   • Area requires splitting at zeros
   • Area is always positive without splitting
   • Area equals derivative of the function

2. Calculate the area bounded by $$y=x^3-2x$$ and $$y=(x+1)^2$$ in the second quadrant.

   • $$2$$
   • $$\frac52$$
   • $$\frac74$$
   • $$\frac{11}{12}-\frac{2\sqrt2}{3}$$

3. Calculate the area in the first quadrant bounded by $$x=4$$ and $$y=x^2+1$$ twice—once with horizontal rectangles and once with vertical ones.

   • $$\frac{76}{3}$$
   • $$12$$
   • $$\frac{32}{3}$$
   • $$16$$

4. Calculate the area bounded by $$x=y^2+2y-3$$ and $$y=-x+2$$.

   • $$\frac92$$
   • $$6$$
   • $$\frac{29\sqrt{29}}{6}$$
   • $$5$$

5. Calculate the area bounded by $$y=2x^3-6x,\ y=5x^2,\ x=0,$$ and $$x=-1$$.

   • $$\frac73$$
   • $$\frac56$$
   • $$\frac52$$
   • $$\frac83$$

6. Find the volume generated by revolving the area bounded by the curves $$y=\sqrt{x+2},\ x=3,$$ and $$y=1$$ about the x-axis.

   • $$2\pi$$
   • $$3\pi$$
   • $$4\pi$$
   • $$8\pi$$

7. Find the volume generated by revolving the area in the first quadrant bounded by $$y=x^2$$ and $$x=2$$ about the line $$x=2$$.

   • $$\frac{8\pi}{3}$$
   • $$\frac{5\pi}{3}$$
   • $$4\pi$$
   • $$\frac{32\pi}{3}$$

8. Find the volume generated by revolving the region bounded by $$y=(x+1)^3,\ y=0,$$ and $$x=0$$ about $$x=2$$.

   • $$\frac{15\pi}{4}$$
   • $$6\pi$$
   • $$\frac{11\pi}{10}$$
   • $$5\pi$$

9. Find the volume generated by revolving the region bounded by $$y=\sin x$$, $$y=0$$, and $$x=\frac{\pi}{3}$$ about the x-axis. Then find $$c$$ such that the volume is split in half.

   • $$c=\frac{\pi}{6}$$
   • $$c=\frac{\pi}{4}$$
   • $$c=\frac{\pi}{5}$$
   • $$c=\frac{\pi}{3}$$

10. What volume results if you rotate the region in the first quadrant bounded by $$y=x^2+2$$, $$x=1$$, and $$y=2$$ about the line $$y=1$$?

    • $$\frac{5\pi}{3}$$
    • $$2\pi$$
    • $$\frac{7\pi}{3}$$
    • $$\frac{13\pi}{15}$$