# 6.6E: Exercises for Section 6.6

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In exercises 1 - 6, calculate the center of mass for the collection of masses given.

1) $$m_1=2$$ at $$x_1=1$$ and $$m_2=4$$ at $$x_2=2$$

2) $$m_1=1$$ at $$x_1=−1$$ and $$m_2=3$$ at $$x_2=2$$

$$x = \frac{5}{4}$$

3) $$m=3$$ at $$x=0,1,2,6$$

4) Unit masses at $$(x,y)=(1,0),(0,1),(1,1)$$

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

5) $$m_1=1$$ at $$(1,0)$$ and $$m_2=4$$ at $$(0,1)$$

6) $$m_1=1$$ at $$(1,0)$$ and $$m_2=3$$ at $$(2,2)$$

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

In exercises 7 - 16, compute the center of mass $$\bar x.$$

7) $$ρ=1$$ for $$x∈(−1,3)$$

8) $$ρ=x^2$$ for $$x∈(0,L)$$

$$\dfrac{3L}{4}$$

9) $$ρ=1$$ for $$x∈(0,1)$$ and $$ρ=2$$ for $$x∈(1,2)$$

10) $$ρ=\sin x$$ for $$x∈(0,π)$$

$$\frac{π}{2}$$

11) $$ρ=\cos x$$ for $$x∈\left(0,\frac{π}{2}\right)$$

12) $$ρ=e^x$$ for $$x∈(0,2)$$

$$\dfrac{e^2+1}{e^2−1}$$

13) $$ρ=x^3+xe^{−x}$$ for $$x∈(0,1)$$

14) $$ρ=x\sin x$$ for $$x∈(0,π)$$

$$\dfrac{π^2−4}{π}$$

15) $$ρ=\sqrt{x}$$ for $$x∈(1,4)$$

16) $$ρ=\ln x$$ for $$x∈(1,e)$$

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

In exercises 17 - 19, compute the center of mass $$(\bar{x},\bar{y}).$$ Use symmetry to help locate the center of mass whenever possible.

17) $$ρ=7$$ in the square $$0≤x≤1, \; 0≤y≤1$$

18) $$ρ=3$$ in the triangle with vertices $$(0,0), \, (a,0)$$, and $$(0,b)$$

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

19) $$ρ=2$$ for the region bounded by $$y=\cos(x), \; y=−\cos(x), \; x=−\frac{π}{2}$$, and $$x=\frac{π}{2}$$

In exercises 20 - 26, 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.

20) [T] The region bounded by $$y=\cos(2x), \; x=−\frac{π}{4}$$, and $$x=\frac{π}{4}$$

$$\left(0,\frac{π}{8}\right)$$

21) [T] The region between $$y=2x^2, \; y=0, \; x=0,$$ and $$x=1$$

22) [T] The region between $$y=\frac{5}{4}x^2$$ and $$y=5$$

$$(0,3)$$

23) [T] Region between $$y=\sqrt{x}, \; y=\ln x, \; x=1,$$ and $$x=4$$

24) [T] The region bounded by $$y=0$$ and $$\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$$

$$\left(0,\frac{4}{π}\right)$$

25) [T] The region bounded by $$y=0, \; x=0,$$ and $$\dfrac{x^2}{4}+\dfrac{y^2}{9}=1$$

26) [T] The region bounded by $$y=x^2$$ and $$y=x^4$$ in the first quadrant

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

In exercises 27 - 31, use the theorem of Pappus to determine the volume of the shape.

27) Rotating $$y=mx$$ around the $$x$$-axis between $$x=0$$ and $$x=1$$

28) Rotating $$y=mx$$ around the $$y$$-axis between $$x=0$$ and $$x=1$$

$$V = \frac{mπ}{3}$$ units³

29) 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?

30) 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?

$$V = πa^2b$$ units³

31) 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 32 - 36, 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.

32) [T] Quarter-circle: $$y=\sqrt{1−x^2}, \; y=0$$, and $$x=0$$

$$\left(\frac{4}{3π},\, \frac{4}{3π}\right)$$

33) [T] Triangle: $$y=x, \; y=2−x$$, and $$y=0$$

34) [T] Lens: $$y=x^2$$ and $$y=x$$

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

35) [T] Ring: $$y^2+x^2=1$$ and $$y^2+x^2=4$$

36) [T] Half-ring: $$y^2+x^2=1, \; y^2+x^2=4,$$ and $$y=0$$

$$\left(0,\, \frac{28}{9π}\right)$$

37) 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.

38) 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.

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

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

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

41) 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.)