Basic Technical Mathematics with Calculus

Allyn J. Washington, Richard S. Evans

Chapter 26

Applications of Integration - all with Video Answers

Educators

Section 4

Centroids

Problem 1

Make the given changes in the indicated examples of this section and then find the coordinates of the centroid. In Example 4, change $y=x^{2}$ to $y=|x|(y=x \text { for } x \geq 0$, and $y=-x$ for $x<0$.

Stanley E.

Stanley E.

Numerade Educator

Problem 2

Make the given changes in the indicated examples of this section and then find the coordinates of the centroid. In Example $6,$ change $y=4-x^{2}$ to $y=4-x$.

Stanley E.

Numerade Educator

Problem 3

Find the center of mass (in cm) of the particles with the given masses located at the given points on the $x$ -axis. $$\text { 5.0 } \mathrm{g} \text { at }(1.0,0), 8.5 \mathrm{g} \text { at }(4.2,0), 3.6 \mathrm{g} \text { at }(7.3,0)$$

Stanley E.

Numerade Educator

Problem 4

Find the center of mass (in cm) of the particles with the given masses located at the given points on the $x$ -axis. $$2.3 \mathrm{g} \text { at }(1.3,0), 6.5 \mathrm{g} \text { at }(5.8,0), 1.2 \mathrm{g} \text { at }(9.5,0)$$

Stanley E.

Numerade Educator

Problem 5

Find the center of mass (in cm) of the particles with the given masses located at the given points on the $x$ -axis. $$\text { 31 } \mathrm{g} \text { at }(-3.5,0), 24 \mathrm{g} \text { at }(0,0), 15 \mathrm{g} \text { at }(2.6,0), 84 \mathrm{g} \text { at }(3.7,0)$$

Stanley E.

Numerade Educator

Problem 6

Find the center of mass (in cm) of the particles with the given masses located at the given points on the $x$ -axis. $$\begin{aligned} 550 \mathrm{g} \text { at }(-42,0), 230 \mathrm{g} \text { at }(-27,0), 470 \mathrm{g} \text { at }(16,0), 120 \mathrm{g} \text { at } (22,0) \end{aligned}$$

Stanley E.

Numerade Educator

Problem 7

Find the coordinates (to 0.01 in. ) of the centroids of the uniform flat-plate machine parts shown.

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Problem 8

Find the coordinates (to 0.01 in. ) of the centroids of the uniform flat-plate machine parts shown.

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Problem 9

Find the coordinates (to 0.01 in. ) of the centroids of the uniform flat-plate machine parts shown.

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Problem 10

Find the coordinates (to 0.01 in. ) of the centroids of the uniform flat-plate machine parts shown.

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Problem 11

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=x^{2}$ and $y=2$

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Problem 12

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The semicircular region in Fig. 26.44.

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Problem 13

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=4-x$ and the axes

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Problem 14

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=x^{3}, x=2,$ and the $x$ -axis

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Problem 15

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=4 x^{2}$ and $y=2 x^{3}$

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Problem 16

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y^{2}=x, y=2,$ and $x=0$

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Problem 17

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=2(x+1), y=3 x+2,$ and $y=8$

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Problem 18

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=x^{2 / 3}, x=8,$ and $y=0$

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Problem 19

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=4-2 x, x=2,$ and $y=4$

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Problem 20

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $y=\sqrt{x}, y=0,$ and $x=9$

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Problem 21

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region bounded by $x^{2}=4 p y$ and $y=a$ if $p>0$ and $a>0$

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Problem 22

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The region above the $x$ -axis, bounded by the ellipse with vertices (a, 0) and $(-a, 0),$ and minor axis $2 b$. (The area of an ellipse is $\pi a b$.

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Problem 23

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by $y=x^{3}, y=0,$ and $x=1$ about the $x$ -axis.

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Problem 24

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by $y=2-2 x, x=0,$ and $y=0$ about the $y$ -axis.

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Problem 25

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region in the first quadrant bounded by $y^{2}=4 x, x=0,$ and $y=2$ about the $y$ -axis.

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Problem 26

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region bounded by $y=x^{2}, x=2,$ and the $x$ -axis about the $x$ -axis.

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Problem 27

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region in the first quadrant bounded by $y^{2}=4 x$ and $x=1$ about the $x$ -axis.

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Problem 28

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region in the first quadrant bounded by $x^{2}-y^{2}=9, y=4,$ and the $x$ -axis about the $y$ -axis.

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Problem 29

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A sailboat has a right-triangular sail with a horizontal base $3.0 \mathrm{m}$ long and a vertical side of $4.5 \mathrm{m}$ high. Where is the centroid of the sail?

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Problem 30

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A lens with semielliptical vertical cross sections and circular horizontal cross sections is shown in Fig. $26.45 .$ For proper installation in an optical device, its centroid must be known. Locate its centroid.

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Problem 31

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. Find the location of the centroid of a hemisphere of radius $a$. Using this result, locate the centroid of the northern hemisphere of Earth for which the radius is $6370 \mathrm{km}$.

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Problem 32

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A sanding machine disc can be described as the solid generated by rotating the region bounded by $y^{2}=4 / x, y=1, y=2,$ and the $y$ -axis about the $y$ -axis (measurements in in.). Locate the centroid of the disc.

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Problem 33

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A highway marking pylon has the shape of a frustum of a cone. Find its centroid if the radii of its bases are $5.00 \mathrm{cm}$ and $20.0 \mathrm{cm}$ and the height between bases is $60.0 \mathrm{cm}$.

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Problem 34

Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. A floodgate is in the shape of an isosceles trapezoid. Find the location of the centroid of the floodgate if the upper base is $20 \mathrm{m}$ the lower base is $12 \mathrm{m},$ and the height between bases is $6.0 \mathrm{m} .$ See Fig. 26.46.

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