Applications of Integration — Quiz 2

Questions in this quiz (10)

1. Suppose the following is a table of coordinates for $$y=f(x)$$, given that $$f$$ is continuous on $$[2,10]$$:If a trapezoidal sum is used with $$n=4$$, then the approximate area under the curve from $$x=2$$ to $$x=10$$, to two decimal places, is:

   • $$47.40$$
   • $$51.80$$
   • $$54.00$$
   • $$55.60$$

2. The region bounded by the curves $$y=4x-x^2$$ and $$y=x$$ is rotated about the $$x$$-axis. Which of the following integrals represents the volume of the solid generated?

   • $$\pi\int_0^3\left[(4x-x^2)^2-x^2\right]dx$$
   • $$\pi\int_0^4\left[(4x-x^2)^2-x^2\right]dx$$
   • $$\pi\int_0^3\left(16x^2-8x^3\right)dx$$
   • $$\pi\int_0^2\left[(4x-x^2)^2-x^2\right]dx$$

3. The region bounded by $$y=\ln(x+1)$$, $$y=0$$, and $$x=e-1$$ is rotated about the line $$x=e$$. Which of the following integrals represents the volume of the solid generated?

   • $$\pi\int_0^1(e-e^y)^2dy$$
   • $$2\pi\int_0^1(e-(e^y-1))y\,dy$$
   • $$2\pi\int_0^{e-1}(e-x)\ln(x+1)dx$$
   • $$\pi\int_0^1(e^2-2ee^y+e^{2y})dy$$

4. If the curves of $$f(x)$$ and $$g(x)$$ intersect for $$x=c$$ and $$x=d$$ and if $$f(x)$$ > $$g(x)$$ > $$0$$ for all $$x$$ on $$(c,d)$$, then the volume obtained when the region bounded by the curves is rotated about the $$y$$-axis is equal to:

   • $$\pi\int_c^d\left[(f(x))^2-(g(x))^2\right]dx$$
   • $$2\pi\int_c^dx[f(x)-g(x)]dx$$
   • $$\pi\int_c^d\left[(f(x))^2-(g(x))^2\right]dx$$
   • $$2\pi\int_c^d[f(x)+g(x)]dx$$

5. A particle moves along a line in such a way that its position at time $$t$$ is given by$$s=t^3-6t^2+11t+2$$Its direction of motion changes when:

   • $$t=\frac{\sqrt{3}}{3}$$ only
   • $$t=3$$ only
   • $$t=2-\frac{\sqrt{3}}{3}$$ and $$t=2+\frac{\sqrt{3}}{3}$$
   • $$t=2$$ only

6. The acceleration of a particle moving on a straight line is given by $$a=\sin t$$ and when $$t=0$$ the particle is at rest. The distance it covers from $$t=0$$ to $$t=2$$ is:

   • $$1-\cos2$$
   • $$2-\sin2$$
   • $$\cos2-1$$
   • $$2+\cos2$$

7. A body moves along a straight line so that its velocity $$v$$ at time $$t$$ is given by$$v=3t^3+4t^2+2$$The distance the body covers from $$t=0$$ to $$t=2$$ equals:

   • $$28$$
   • $$30$$
   • $$34$$
   • $$\frac{80}{3}$$

8. During a certain $$3$$-hour period the wind velocity, in miles per hour, is given by$$v(t)=4t^2-6t+80,\ 0\le t\le3$$The average wind velocity during this period (in mph) is:

   • $$83$$
   • $$84$$
   • $$86$$
   • $$88$$

9. A particle moves along a line with velocity$$v(t)=2t^2-4t$$The net change in position of the particle from $$t=0$$ to $$t=3$$ is:

   • $$0$$
   • $$3$$
   • $$4$$
   • $$6$$

10. During a certain $$5$$-hour period of a storm the wind velocity, in miles per hour, is given by$$v(t)=6t-t^2+90,\ 0\le t\le5$$The average wind velocity during this period (in mph) is:

    • $$\frac{290}{3}$$
    • $$101\frac{2}{3}$$
    • $$103\frac{1}{3}$$
    • $$105$$