1.7E: Exercises

In exercises 1 - 4, find the weight of the one-dimensional object.

1) A wire that is \(2\) ft long (starting at \(x=0\)) and has a weight density function of \(\gamma(x)=x^2+2x\) lb/ft

2) A car antenna that is \(3\) ft long (starting at \(x=0)\) and has a weight density function of \(\gamma(x)=3x+2\) lb/ft

Answer

\( \frac{39}{2}\) lbs.

3) A metal rod that is \( 8\) in. long (starting at \( x=0\)) and has a weight density function of \( \gamma(x)=e^{1/2x}\) lb/in.

4) A pencil that is \( 4\) in. long (starting at \( x=2\)) and has a weight density function of \( \gamma(x)=\dfrac{5}{x}\) oz/in.

Answer

\( \ln(243)\) oz.

5) Using the Left Endpoint Method with \( n = 6 \), approximate the weight of a one-dimensional ruler that is \( 12\) in. long (starting at \( x=5\)) and has a weight density function of \( \gamma(x)=\ln(x)+(1/2)x^2\) oz/in.

In exercises 6 - 7, find the weight of the two-dimensional object that is centered at the origin.

6) An oversized hockey puck of radius \( 2\) in. with radial weight density function \( \gamma(x)=x^3−2x+5\) \( \text{lb}/\text{in}^2 \).

Answer

\( \frac{332\pi }{15}\) lbs.

7) A disk of radius \(5\) cm with radial weight density function \(\gamma(x)=\sqrt{3x}\) \( \text{g}/\text{cm}^2 \)

Answer

\(20\pi \sqrt{15}\) g.

8) Using the Right Endpoint Method with \( n = 6 \), approximate the weight of a two-dimensional frisbee (centered at the origin) of radius \( 6\) in. with radial weight density function \( \gamma(x)=e^{−x}\) \( \text{oz}/\text{in}^2 \).

In exercises 9 - 14, calculate the center of mass for the collection of masses given.

9) \(m_1=2\) at \(x_1=1\) and \(m_2=4\) at \(x_2=2\)

10) \(m_1=1\) at \(x_1=−1\) and \(m_2=3\) at \(x_2=2\)

Answer

\(x = \frac{5}{4}\)

11) \(m=3\) at \(x=0,1,2,6\)

12) Unit masses at \((x,y)=(1,0),(0,1),(1,1)\)

Answer

\(\left(\frac{2}{3},\, \frac{2}{3}\right)\)

13) \(m_1=1\) at \((1,0)\) and \(m_2=4\) at \((0,1)\)

14) \(m_1=1\) at \((1,0)\) and \(m_2=3\) at \((2,2)\)

Answer

\(\left(\frac{7}{4},\,\frac{3}{2}\right)\)

In exercises 15 - 24, compute the center of mass \(\bar x.\)

15) \( \rho =1\) for \(x \in (−1,3)\)

16) \( \rho =x^2\) for \(x \in (0,L)\)

Answer

\(\dfrac{3L}{4}\)

17) \( \rho =1\) for \(x \in (0,1)\) and \( \rho =2\) for \(x \in (1,2)\)

18) \( \rho =\sin x\) for \(x \in (0, \pi )\)

Answer

\(\frac{ \pi }{2}\)

19) \( \rho =\cos x\) for \(x \in \left(0,\frac{ \pi }{2}\right)\)

20) \( \rho =e^x\) for \(x \in (0,2)\)

Answer

\(\dfrac{e^2+1}{e^2−1}\)

21) \( \rho =x^3+xe^{−x}\) for \(x \in (0,1)\)

22) \( \rho =x\sin x\) for \(x \in (0, \pi )\)

Answer

\(\dfrac{ \pi ^2−4}{ \pi }\)

23) \( \rho =\sqrt{x}\) for \(x \in (1,4)\)

24) \( \rho =\ln x\) for \(x \in (1,e)\)

Answer

\(\frac{1}{4}(1+e^2)\)

In exercises 25 - 27, compute the center of mass \((\bar{x},\bar{y}).\) Use symmetry to help locate the center of mass whenever possible.

25) \( \rho =7\) in the square \(0 \leq x \leq 1, \; 0 \leq y \leq 1\)

26) \( \rho =3\) in the triangle with vertices \((0,0), \, (a,0)\), and \((0,b)\)

Answer

\(\left(\frac{a}{3},\, \frac{b}{3}\right)\)

27) \( \rho =2\) for the region bounded by \(y=\cos(x), \; y=−\cos(x), \; x=−\frac{ \pi }{2}\), and \(x=\frac{ \pi }{2}\)

In exercises 28 - 34, use a calculator to draw the region, then compute the center of mass \((\bar{x},\bar{y}).\) Use symmetry to help locate the center of mass whenever possible.

28) [Technology Required] The region bounded by \(y=\cos(2x), \; x=−\frac{ \pi }{4}\), and \(x=\frac{ \pi }{4}\)

Answer

\(\left(0,\frac{ \pi }{8}\right)\)

29) [Technology Required] The region between \(y=2x^2, \; y=0, \; x=0,\) and \(x=1\)

30) [Technology Required] The region between \(y=\frac{5}{4}x^2\) and \(y=5\)

Answer

\((0,3)\)

31) [Technology Required] Region between \(y=\sqrt{x}, \; y=\ln x, \; x=1,\) and \(x=4\)

32) [Technology Required] The region bounded by \(y=0\) and \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)

Answer

\(\left(0,\frac{4}{ \pi }\right)\)

33) [Technology Required] The region bounded by \(y=0, \; x=0,\) and \(\dfrac{x^2}{4}+\dfrac{y^2}{9}=1\)

34) [Technology Required] The region bounded by \(y=x^2\) and \(y=x^4\) in the first quadrant

Answer

\(\left(\frac{5}{8},\, \frac{1}{3}\right)\)

In exercises 35 - 39, use the Theorem of Pappus to determine the volume of the shape.

35) Rotating \(y=mx\) around the \(x\)-axis between \(x=0\) and \(x=1\)

36) Rotating \(y=mx\) around the \(y\)-axis between \(x=0\) and \(x=1\)

Answer

\(V = \frac{m \pi }{3}\) units³

37) A general cone created by rotating a triangle with vertices \((0,0), \, (a,0),\) and \((0,b)\) around the \(y\)-axis. Does your answer agree with the volume of a cone?

38) A general cylinder created by rotating a rectangle with vertices \((0,0), \, (a,0), \, (0,b),\) and \((a,b)\) around the \(y\)-axis. Does your answer agree with the volume of a cylinder?

Answer

\(V = \pi a^2b\) units³

39) A sphere created by rotating a semicircle with radius \(a\) around the \(y\)-axis. Does your answer agree with the volume of a sphere?

In exercises 40 - 44, use a calculator to draw the region enclosed by the curve. Find the area \(M\) and the centroid \((\bar{x},\bar{y})\) for the given shapes. Use symmetry to help locate the center of mass whenever possible.

40) [Technology Required] Quarter-circle: \(y=\sqrt{1−x^2}, \; y=0\), and \(x=0\)

Answer

\(\left(\frac{4}{3 \pi },\, \frac{4}{3 \pi }\right)\)

41) [Technology Required] Triangle: \(y=x, \; y=2−x\), and \(y=0\)

42) [Technology Required] Lens: \(y=x^2\) and \(y=x\)

Answer

\(\left(\frac{1}{2},\, \frac{2}{5}\right)\)

43) [Technology Required] Ring: \(y^2+x^2=1\) and \(y^2+x^2=4\)

44) [Technology Required] Half-ring: \(y^2+x^2=1, \; y^2+x^2=4,\) and \(y=0\)

Answer

\(\left(0,\, \frac{28}{9 \pi }\right)\)

45) Find the generalized center of mass in the sliver between \(y=x^a\) and \(y=x^b\) with \(a>b\). Then, use the Pappus theorem to find the volume of the solid generated when revolving around the \(y\)-axis.

46) Find the generalized center of mass between \(y=a^2−x^2, \; x=0\), and \(y=0\). Then, use the Pappus theorem to find the volume of the solid generated when revolving around the \(y\)-axis.

Answer

Center of mass: \(\left(\frac{a}{6},\,\frac{4a^2}{5}\right),\)
Volume: \(\dfrac{2 \pi a^4}{9}\) units³

47) Find the generalized center of mass between \(y=b\sin(ax),\; x=0,\) and \(x=\dfrac{ \pi }{a}.\) Then, use the Theorem of Pappus to find the volume of the solid generated when revolving around the \(y\)-axis.

48) Use the Theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius \(a\) is positioned with the left end of the circle at \(x=b, \, b>0,\) and is rotated around the \(y\)-axis.

Answer

Volume: \(V = 2\pi^2a^2(b+a)\)

49) Find the center of mass \((\bar{x},\bar{y})\) for a thin wire along the semicircle \(y=\sqrt{1−x^2}\) with unit mass. (Hint: Use the Theorem of Pappus.)

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.