Centroids by integration

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Question statement

{applet}

Use integration with {if(Vertical=1,'vertical','horizontal')} strips to determine the area of the spandrel bounded by the function $\color{red}{y = f(x) = h - k \simplify[all,fractionnumbers]{x^{n}}}$ and $\var{if(Above=1, 'the lines $y=h$ and $x=b$','the $x$- and $y$-axes')}$ , and also find the coordinates of its centroid. Note: b and h are constants.

Give any introductory information the student needs.

			Name	Type	Generated Value

n	rational	1/2
version	integer	3
Vertical	integer	1
Above	integer	1
k	expression	h/b^n
debug	boolean	false

			Name	Type	Generated Value

dA	expression	(h - y)dx
xbar_el	expression	x
ybar_el	expression	(h + y)/2
upper_limit	string	b
lower_limit	integer	0
area	expression	b*h/(3/2)
Qy	expression	h*b^2/(5/2)
xbar	expression	b*(3/2)/(5/2)
ybar	expression	h*(5/2)/4
Qx	expression	(5/2)(b*h^2/2)/3

			Name	Type	Generated Value

applet

ggbapplet

HTML node

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1/3,1/2,1,3/2,2,3)

JME function reference

Description

the exponent of the function (1-kx^n) (0.5 .. 5#0.5)

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `rational`

1/2

→ Used by:

Qx
Qy
applet
area
xbar
ybar

This variable doesn't seem to be used anywhere.

Parts

Bounding function Gap-fill
1. Gap k Mathematical expression
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Strip Gap-fill
Limits Gap-fill
1. Gap lower Match text pattern
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2. Gap upper Match text pattern
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Area Gap-fill
1. Gap area Mathematical expression
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First Moments Gap-fill
1. Gap qx Mathematical expression
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2. Gap qy Mathematical expression
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Centroid Gap-fill
1. Gap xbar Mathematical expression
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2. Gap ybar Mathematical expression
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Ask the student a question, and give any hints about how they should answer this part.

The bounding funtion contains a constant $k$ which can be expressed in terms of $b$ and $h$. Determine the value of $k$ in terms of b and h by substituting in the coordinates of a point which is on the curve into the bounding function.

$k$ = k {k}

Advice

Note: The solutions below is for Vertical Strips.

The procedure for horizontal strips is similar, but with these strip properties and limits:

Strip properties: $dA = (b-x)\,dy$, $\bar{x}_{el} = \dfrac{x+b}{y}$, $\bar{y}_{el} = y $ $dA = x \,dy$, $\bar{x}_{el} = x/2$, $\bar{y}_{el} =y$

Limits: $x= 0 \text{ to } h$

You will get the same results for $A, Q_x, Q_y, \bar{x} \text{ and } \bar{y}$, which ever strips you choose.

1. Bounding function: $y = f(x) = h - k \simplify[all,fractionnumbers]{x^{n}}$

2. Constant $k$ in terms of $b$: at $x=b, y=0 \therefore h - k \simplify[all,fractionnumbers]{b^{n}} = 0 \therefore k = \dfrac{h}{\simplify[all,fractionnumbers]{b^{n}}}$

3. Strip properties: $dA = y\,dx$, $\bar{x}_{el} = x$, $\bar{y}_{el} = y/2$ $dA = (h-y)\,dx$, $\bar{x}_{el} = x$, $\bar{y}_{el} = (h+y)/2$

4. Limits: $x= 0 \text{ to } b$

5. Area under the curve:

$\begin{align} A &= \int dA\\&= \int_0^b y\, dx\\&= \int_0^b (h - k \simplify[all,fractionnumbers]{x^{n}})\, dx\\&= \left[ hx - k \dfrac{\simplify[all,fractionnumbers]{x^{n+1}}}{\simplify[all,fractionnumbers]{{n+1}}}\right]_0^b\\&= \left[ hb - \left(\dfrac{h}{\simplify[all,fractionnumbers]{b^{n}}}\right) \dfrac{\simplify[all,fractionnumbers]{b^{n+1}}}{\simplify[all,fractionnumbers]{{n+1}}}\right]\\&=hb-\dfrac{hb}{\simplify[all,fractionnumbers]{{n}+1}}\\&=\var[all,fractionnumbers]{{n}/({n}+1)} h b\end{align}$

$\begin{align} A &= \int dA\\&= \int_0^b (h-y)\, dx\\&= \int_0^b h - (h - k \simplify[all,fractionnumbers]{x^{n}})\, dx\\&=\int_0^bk \simplify[all,fractionnumbers]{x^{n}}dx\\&= \left[k \dfrac{\simplify[all,fractionnumbers]{x^{n+1}}}{\simplify[all,fractionnumbers]{{n+1}}}\right]_0^b\\&= \left[ \left(\dfrac{h}{\simplify[all,fractionnumbers]{b^{n}}}\right) \dfrac{\simplify[all,fractionnumbers]{b^{n+1}}}{\simplify[all,fractionnumbers]{{n+1}}}\right]\\&=\dfrac{hb}{\simplify[all,fractionnumbers]{{n}+1}}\\&=\var[all,fractionnumbers]{1/({n}+1)} h b\end{align}$

6. $Q_x$ and $Q_y$ Perform similar integrations to get:

$Q_x = \int \bar{y}_{el}\,dA = \simplify[all,simplifyfractions,fractionnumbers]{{n}^2 /(2{n}^2+3{n}+1) h^2b} \qquad Q_y = \int \bar{x}_{el}\,dA = \simplify[all,fractionnumbers,simplifyfractions]{{n}/(2{n}+4)} hb^2$

$Q_x = \int \bar{y}_{el}\,dA = \simplify[all,simplifyfractions,fractionnumbers]{(3{n}+1)/(2({n}+1)(2{n}+1))}h^2b \qquad Q_y = \int \bar{x}_{el}\,dA = \simplify[all,fractionnumbers,simplifyfractions]{1/({n}+2)} hb^2$

7. Centroid:

$\bar{x} = \dfrac{Q_y}{A} = \dfrac{\simplify[all,fractionnumbers,simplifyfractions]{{n}/(2{n}+4)} hb^2}{\var[all,fractionnumbers]{{n}/({n}+1)} h b} = \simplify[all,collectnumbers]{({n}+1)/(2{n}+4)}b \qquad \bar{y} = \dfrac{Q_x}{A} = \dfrac{\simplify[all,fractionnumbers,simplifyfractions]{{n^2}/(({n}+1)(2{n}+1))} h^2b}{\var[all,fractionnumbers]{{n}/({n}+1)} h b} = \simplify[all,collectnumbers,simplifyfractions]{{n}/(2{n}+1)}h $

$\bar{x} = \dfrac{Q_y}{A} = \dfrac{\simplify[all,fractionnumbers,simplifyfractions]{1/({n}+2)} hb^2}{\var[all,fractionnumbers]{1/({n}+1)} h b} = \simplify[all,collectnumbers]{({n}+1)/({n}+2)}b \qquad \bar{y} = \dfrac{Q_x}{A} = \dfrac{\simplify[all,simplifyfractions,fractionnumbers]{(3{n}+1)/(2({n}+1)(2{n}+1))}h^2b}{\var[all,fractionnumbers]{1/({n}+1)} h b} = \simplify[all,collectnumbers,simplifyfractions]{(3{n}+1)/(4{n}+2)}h $

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This question is used in the following exams:

Exam 2 Practice Fall 2020 by William Haynes in Online Exams.
Chapter 7 Exercises by William Haynes in Engineering Statics.
Chapter 07 Exercises by Michael Proudman in Engineering Statics.
20. Centroids by Integration by Michael Proudman in Engineering Statics.
21. Centroids by Integration by William Haynes in Engineering Statics.

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Centroids by integration

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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William Haynes 2 weeks, 3 days ago

William Haynes 2 weeks, 3 days ago

William Haynes 4 years, 4 months ago

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William Haynes 5 years, 11 months ago

William Haynes 6 years ago

William Haynes 6 years ago

William Haynes 6 years, 7 months ago

William Haynes 6 years, 7 months ago

William Haynes 6 years, 7 months ago

Error in variable testing condition

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Generated value: rational

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Generated value: `rational`