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# 1.5E: Exercises

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## Moments and Centers of Mass

For the following exercises, calculate the center of mass for the collection of masses given.

Exercise $$\PageIndex{1}$$

$$\displaystyle m_1=2$$ at $$\displaystyle x_1=1$$ and $$\displaystyle m_2=4$$ at $$\displaystyle x_2=2$$

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Exercise $$\PageIndex{2}$$

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

$$\displaystyle \frac{5}{4}$$

Exercise $$\PageIndex{3}$$

$$\displaystyle m=3$$ at $$\displaystyle x=0,1,2,6$$

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Exercise $$\PageIndex{4}$$

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

$$\displaystyle (\frac{2}{3},\frac{2}{3})$$

Exercise $$\PageIndex{5}$$

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

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Exercise $$\PageIndex{6}$$

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

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

For the following exercises, compute the center of mass x–.

Exercise $$\PageIndex{7}$$

$$\displaystyle ρ=1$$ for $$\displaystyle x∈(−1,3)$$

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Exercise $$\PageIndex{8}$$

$$\displaystyle ρ=x^2$$ for $$\displaystyle x∈(0,L)$$

$$\displaystyle \frac{3L}{4}$$

Exercise $$\PageIndex{9}$$

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

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Exercise $$\PageIndex{10}$$

$$\displaystyle ρ=sinx$$ for $$\displaystyle x∈(0,π)$$

$$\displaystyle \frac{π}{2}$$

Exercise $$\PageIndex{11}$$

$$\displaystyle ρ=cosx$$ for $$\displaystyle x∈(0,\frac{π}{2})$$

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Exercise $$\PageIndex{12}$$

$$\displaystyle ρ=e^x$$ for $$\displaystyle x∈(0,2)$$

$$\displaystyle \frac{e^2+1}{e^2−1}$$

Exercise $$\PageIndex{13}$$

$$\displaystyle ρ=x^3+xe^{−x}$$ for $$\displaystyle x∈(0,1)$$

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Exercise $$\PageIndex{14}$$

$$\displaystyle ρ=xsinx$$ for $$\displaystyle x∈(0,π)$$

$$\displaystyle \frac{π^2−4}{π}$$

Exercise $$\PageIndex{15}$$

$$\displaystyle ρ=\sqrt{x}$$ for $$\displaystyle x∈(1,4)$$

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Exercise $$\PageIndex{16}$$

$$\displaystyle ρ=lnx$$ for $$\displaystyle x∈(1,e)$$

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

For the following exercises, compute the center of mass $$\displaystyle (\bar{x},\bar{y})$$. Use symmetry to help locate the center of mass whenever possible.

Exercise $$\PageIndex{17}$$

$$\displaystyle ρ=7$$ in the square $$\displaystyle 0≤x≤1, 0≤y≤1$$

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Exercise $$\PageIndex{18}$$

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

$$\displaystyle (\frac{a}{3},\frac{b}{3})$$

Exercise $$\PageIndex{19}$$

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

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For the following exercises, use a calculator to draw the region, then compute the center of mass $$\displaystyle (\bar{x},\bar{y})$$. Use symmetry to help locate the center of mass whenever possible.

Exercise $$\PageIndex{20}$$

The region bounded by $$\displaystyle y=cos(2x), x=−\frac{π}{4}$$, and $$\displaystyle x=\frac{π}{4}$$

$$\displaystyle (0,\frac{π}{8})$$

Exercise $$\PageIndex{21}$$

The region between $$\displaystyle y=2x^2, y=0, x=0,$$ and $$\displaystyle x=1$$

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Exercise $$\PageIndex{22}$$

The region between $$\displaystyle y=\frac{5}{4}x^2$$ and $$\displaystyle y=5$$

$$\displaystyle (0,3)$$

Exercise $$\PageIndex{23}$$

Region between $$\displaystyle y=\sqrt{x}, y=ln(x), x=1,$$ and $$\displaystyle x=4$$

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Exercise $$\PageIndex{24}$$

The region bounded by $$\displaystyle y=0, \frac{x^2}{4}+\frac{y^2}{9}=1$$

$$\displaystyle (0,\frac{4}{π})$$

Exercise $$\PageIndex{25}$$

The region bounded by $$\displaystyle y=0, x=0,$$ and $$\displaystyle \frac{x^2}{4}+\frac{y^2}{9}=1$$

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Exercise $$\PageIndex{26}$$

The region bounded by $$\displaystyle y=x^2$$ and $$\displaystyle y=x^4$$ in the first quadrant

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

For the following exercises, use the theorem of Pappus to determine the volume of the shape.

Exercise $$\PageIndex{27}$$

Rotating $$\displaystyle y=mx$$ around the $$\displaystyle x$$-axis between $$\displaystyle x=0$$ and $$\displaystyle x=1$$

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Exercise $$\PageIndex{28}$$

Rotating $$\displaystyle y=mx$$ around the $$\displaystyle y$$-axis between $$\displaystyle x=0$$ and $$\displaystyle x=1$$

$$\displaystyle \frac{mπ}{3}$$

Exercise $$\PageIndex{29}$$

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

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Exercise $$\PageIndex{30}$$

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

$$\displaystyle πa^2b$$

Exercise $$\PageIndex{31}$$

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

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

Exercise $$\PageIndex{32}$$

Quarter-circle: $$\displaystyle y=\sqrt{1−x^2}, y=0$$, and $$\displaystyle x=0$$

$$\displaystyle (\frac{4}{3π},\frac{4}{3π})$$

Exercise $$\PageIndex{33}$$

Triangle: $$\displaystyle y=x, y=2−x$$, and $$\displaystyle y=0$$

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Exercise $$\PageIndex{34}$$

Lens: $$\displaystyle y=x^2$$ and $$\displaystyle y=x$$

$$\displaystyle (\frac{1}{2},\frac{2}{5})$$

Exercise $$\PageIndex{35}$$

Ring: $$\displaystyle y^2+x^2=1$$ and $$\displaystyle y^2+x^2=4$$

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Exercise $$\PageIndex{36}$$

Half-ring: $$\displaystyle y^2+x^2=1, y^2+x^2=4,$$ and $$\displaystyle y=0$$

$$\displaystyle (0,\frac{28}{9π})$$

Exercise $$\PageIndex{37}$$

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

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Exercise $$\PageIndex{38}$$

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

Center of mass: $$\displaystyle (\frac{a}{6},\frac{4a^2}{5}),$$ volume: $$\displaystyle \frac{2πa^4}{9}$$

Exercise $$\PageIndex{39}$$

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

Exercise $$\PageIndex{40}$$
Use the theorem of Pappus to find the volume of a torus (pictured here). Assume that a disk of radius $$\displaystyle a$$ is positioned with the left end of the circle at $$\displaystyle x=b, b>0,$$ and is rotated around the y-axis. Find the center of mass $$\displaystyle (\bar{x},\bar{y})$$ for a thin wire along the semicircle $$\displaystyle y=\sqrt{1−x^2}$$ with unit mass. (Hint: Use the theorem of Pappus.)