Applications of Integration — Quiz 1

Questions in this quiz (10)

1. Define area of the region whose boundaries are given by the curve:$$y=x^3$$, $$y=0$$, $$x=-1$$, and $$x=2$$.

   • $$\frac{15}{4}$$
   • $$\frac{17}{4}$$
   • $$3$$
   • $$\frac{7}{2}$$

2. The parabolas $$x=y^2-5y$$ and $$x=3y-y^2$$ enclose a region. Find the area of the region.

   • $$\frac{32}{3}$$
   • $$\frac{139}{6}$$
   • $$\frac{64}{3}$$
   • $$\frac{128}{3}$$

3. The area of the region whose boundaries is defined by the curve $$y=\frac{6}{x^2+9}$$, the $$x$$-axis, and the vertical lines $$x=-3$$ and $$x=3$$ is:

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

4. The total area bounded by the cubic $$x=y^3-y$$ and the line $$x=3y$$ is equal to:

   • $$4$$
   • $$\frac{16}{3}$$
   • $$8$$
   • $$16$$

5. The figure below shows part of the curve $$y=x^2$$ and a rectangle with two vertices at $$(0,0)$$ and $$(a,0)$$. The upper right corner of the rectangle lies on the curve. What is the ratio of the area of the rectangle to the area of the region inside the rectangle above the parabola?

   • $$3:2$$
   • $$2:1$$
   • $$4:3$$
   • $$3:1$$

6. The area bounded by $$y=\ln x$$, $$y=1$$ and the $$x$$-axis is equal to:

   • $$e-1$$
   • $$e-2$$
   • $$1-\ln2$$
   • $$e-3$$

7. The region bounded by $$y=x^2$$ and $$y=9$$ is rotated about the line $$y=9$$. Which of the following gives the volume of the solid generated?

   • $$\frac{432\pi}{5}$$
   • $$\frac{486\pi}{5}$$
   • $$\frac{864\pi}{5}$$
   • $$\frac{1296\pi}{5}$$

8. A trapezoid has vertices at $$(1,0),(1,3),(3,0),(3,5)$$. The region is rotated about the $$x$$-axis. Which of the following is the volume of the solid generated?

   • $$\frac{116\pi}{3}$$
   • $$\frac{134\pi}{3}$$
   • $$\frac{152\pi}{3}$$
   • $$\frac{98\pi}{3}$$

9. The base of a solid is the region inside the circle $$x^2+y^2=b^2$$ centered at the origin. Each cross section perpendicular to the $$x$$-axis is an equilateral triangle whose base lies in the circular region. The volume of the solid is:

   • $$\frac{\sqrt{3}}{2}b^3$$
   • $$\frac{2\sqrt{3}}{3}b^3$$
   • $$\frac{4\sqrt{3}}{3}b^3$$
   • $$\frac{8\sqrt{3}}{3}b^3$$

10. The base of a solid is the region bounded by $$y=e^{-2x}$$, the $$x$$-axis, the $$y$$-axis, and the line $$x=1$$. Each cross section perpendicular to the $$x$$-axis is a square. The volume of the solid is:Each cross section is a square, so its side length is$$e^{-2x}$$Therefore its area is$$A(x)=\left(e^{-2x}\right)^2=e^{-4x}$$The volume is$$V=\int_0^1e^{-4x}dx$$Integrate:$$V=\left[\frac{e^{-4x}}{-4}\right]_0^1$$$$V=-\frac{e^{-4}}{4}-\left(-\frac{1}{4}\right)$$$$V=\frac{1-e^{-4}}{4}$$Final answer: $$\frac{1-e^{-4}}{4}$$

    • $$\frac{1-e^{-4}}{2}$$
    • $$\frac{1-e^{-4}}{4}$$
    • $$\frac{e^4-1}{4}$$
    • $$1-e^{-4}$$