// Numbas version: exam_results_page_options {"name": "21. Centroids by Integration", "metadata": {"description": "

Homework set. Use integration to find the centroid of shapes bounded by functions.

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Given a random spandrel, find the expressions for the differential elements of area and the coordinates of its centroid needed to determine the location of the centroid by integration.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

{geogebra_applet('tpndhsyt',[['version',version]])}

", "advice": "

The area of the strip, dA is the width times the height of the strip.

The narrow width of the element is either $dx$ or $dy$

The coordinates of the centroid are found by averaging the coordinates of the top and bottom of the strip.

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Determine the properties of the differential strip in terms of the known coordinates.

$dA$ = [[0]]

$\\bar{x}_{el}$ = [[1]]

$\\bar{y}_{el}$ =[[2]]

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Student calculates $\\bar{y}$ for a triangle. Must use similar triangles get element $dA$.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

{applet}

", "advice": "

Use similar triangles to express the width of the strip as a function of $y$:

$\\dfrac{w}{h-y} = \\dfrac{b}{h}$

$w = b\\dfrac{(h-y)}{h}$

Therefore,

$dA = w\\; dy = b\\dfrac{(h-y)}{h} dy$

With this you can evaluate $Q_x$

$\\begin{align} Q_x &= \\int y\\; dA\\\\ &=\\int_0^h y\\;b\\frac{(h-y)}{h}\\; dy\\\\&= \\frac{b}{h}\\int_0^h (hy - y^2)\\; dy\\\\& = \\frac{b}{h}\\left. \\left(\\frac{h y^2}{2} - \\frac{y^3}{3}\\right)\\right |_0^h \\\\ &= bh^2\\left( \\frac{1}{2} - \\frac{1}{3}\\right)\\\\ &= \\frac{bh^2}{6}\\end{align}$

Finaly, find $\\bar{y}$ knowing the area of the triangle is $\\dfrac{bh}{2}$

$\\begin{align} \\bar{y} &= \\frac{Q_x}{A}\\\\ &= \\frac{{bh^2}/{6}}{{bh}/{2}}\\\\ &= \\frac{h}{3}\\end{align}$

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Use horizontal strips to prove by integration that for a triangle with base $b$ and height $h$, $\\bar{y} = h/3$.

1. Derive expressions for $\\bar{y}_{el}$ and $dA$ in terms of $y$.

$\\bar{y}_{el} =$ [[1]] $\\qquad dA=$ [[0]]

2. Determine the limits of the integration.

Lower limit [[2]] Upper limit [[3]]

3. Evaluate $Q_x $ between your limits.

$Q_x = \\int \\bar{y}_{el} dA = $ [[4]]

4. Evaluate $\\bar{y}$ for the triangle.

$\\bar{y} = \\dfrac{Q_x}{A} = $ [[5]]

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Find the area, first moment of area, and coordinates of a general spandrel. The area may be above or below the function.

", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "

{geogebra_applet('bqnqgpfp',[['n',n],['h',random(1,1.5)],['b',random(0.8,1.3)],['Above', above],['Vertical',vertical]])}

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.

", "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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expression(\"(\"+ string(Qy) +\")/(\"+ string(area) +\")\")

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area above or below the curve?

", "templateType": "anything", "can_override": false}, "version": {"name": "version", "group": "inputs", "definition": "2 above + vertical\n", "description": "

0 = below horizontal

1 = below vertical

2 = above horizontal

3 = above vertical

horizontal strips not implemented

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0 = horizontal strips

1 = vertical strips? Horizontal strips not currently implemented.

Can't figure out how to pass a boolean to geogebra, so using {0,1} instead. Only exponents < 2 can be horizontal

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

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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.

$k$ = [[0]] {k}

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Determine the area of the rectangular differential strip.

$dA$ = [[0]] {da}

Determine expressions for the coordinates of the centroid of the strip.

$\\bar{x}_{el} = $ [[1]] $\\qquad\\bar{y}_{el} = $ [[2]] xbar= {xbar_el} ybar = {ybar_el}

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Determine the limits of integration for these strips.

Lower limit is: [[0]] {0} $\\qquad$ Upper limit is: [[1]] {upper_limit}

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With the information determined above, evaluate this integral to find the area of the spandrel in terms of constants b and h.

$A = \\int dA = $ [[0]]

area = {area}

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Perform the following integrals to determine the first moments of area with respect to the x- and y-axis.

$Q_x = \\int \\bar{y}_{el} dA =$ [[0]] {Qx} $\\qquad Q_y = \\int \\bar{x}_{el} dA =$ [[1]] {Qy}

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Use the information you found above to determine the coordinates of the centroid of the shaded spandrel.

$\\bar{x} = \\dfrac{Q_y}{A} = $ [[0]] {xbar} $\\qquad\\bar{y} = \\dfrac{Q_x}{A} = $ [[1]] {ybar}

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

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Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

", "definition": "replace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")"}, {"name": "good_units", "description": "", "definition": "try(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n"}, {"name": "student_quantity", "description": "

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

", "definition": "switch(not good_units, \n student_scalar * correct_units, \n not right_sign,\n -quantity(student_scalar, student_units),\n quantity(student_scalar,student_units)\n)\n \n"}, {"name": "percent_error", "description": "", "definition": "try(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n "}, {"name": "right", "description": "", "definition": "percent_error <= settings['right']\n"}, {"name": "close", "description": "

Only marked close if the student actually has the right sign.

", "definition": "right_sign and percent_error <= settings['close']"}, {"name": "right_sign", "description": "", "definition": "sign(student_scalar) = sign(correct_quantity) "}], "settings": [{"name": "correctAnswer", "label": "Correct Quantity.", "help_url": "", "hint": "The correct answer given as a JME quantity.", "input_type": "code", "default_value": "", "evaluate": true}, {"name": "right", "label": "% Accuracy for right.", "help_url": "", "hint": "Question will be considered correct if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "0.2", "evaluate": true}, {"name": "close", "label": "% Accuracy for close.", "help_url": "", "hint": "Question will be considered close if the scalar part of the student's answer is within this % of correct value.", "input_type": "code", "default_value": "1.0", "evaluate": true}, {"name": "C1", "label": "Close with units.", "help_url": "", "hint": "Partial Credit for close value with appropriate units. if correct answer is 100 N and close is ±1%,
99 N is accepted.", "input_type": "percent", "default_value": "75"}, {"name": "C2", "label": "No units or wrong sign", "help_url": "", "hint": "Partial credit for forgetting units or using wrong sign.
If the correct answer is 100 N, both 100 and -100 N are accepted.", "input_type": "percent", "default_value": "50"}, {"name": "C3", "label": "Close, no units.", "help_url": "", "hint": "Partial Credit for close value but forgotten units.
This value would be close if the expected units were provided. If the correct answer is 100 N, and close is ±1%,
99 is accepted.", "input_type": "percent", "default_value": "25"}], "public_availability": "always", "published": true, "extensions": ["quantities"]}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["centroid", "integration", "Mechanics", "mechanics", "Statics", "statics"], "metadata": {"description": "

Use integration to find the centroid of an area bounded by a parabola, a sloping line, and the y-axis.

", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "

{applet()}

Determine the coordinates of the centroid of the shaded area bounded by the $y$-axis, a parabola passing throught the origin and point $A$, and a straight line passing through points $A$ and $B$.

Given: $A = ( \\var{A[0]}, \\var{A[1]} ), B = (0, \\var{b}) $

", "advice": "

Bounding Functions

Parabola: $f(x) = k x^2$

at $x=\\var{A[0]}, f(x)=\\var{A[1]}$ , so $k = \\dfrac{\\var{A[1]}}{\\var{A[0]}^2}$

so, $f(x) = \\var[fractionNumbers]{A[1]/A[0]^2} x^2$

Line: $g(x) = m x + b$

where $m = \\dfrac{(\\var{A[1]}-\\var{b})}{\\var{x}}$, and $b = \\var{b}$

so, g(x) = $\\var[fractionNumbers]{(A[1]-b) /x} x + \\var{b}$

Strip Properties for Vertical strips

$dA = [g(x) - f(x)]\\; dx =\\left(\\simplify[all]{- {A[1]}/{A[0]}^2} x^2 +\\simplify{({A[1]}-{b}) /{x}} x + \\simplify{{b}} \\right )\\; dx $

$\\bar{x}_{el} = x$

$\\bar{y}_{el} = \\dfrac{g(x) + f(x)}{2} $

Integrals

$A = \\int_0^\\var{x}\\; dA = \\var{siground(area,4)} \\text{ units}^2$

$Q_y = \\int_0^\\var{x} \\bar{x}_{el} \\; dA = \\var{siground(Qy,4)} \\text{ units}^3$

$Q_x = \\int_0^\\var{x} \\bar{y}_{el} \\; dA = \\var{siground(Qx,4)}\\text{ units}^3$

Centroid

$\\bar{x} = \\dfrac{Q_y}{A} = \\var{siground(xbar,4)}$ units

$\\bar{y} = \\dfrac{Q_x}{A} = \\var{siground(ybar,4)}$ units

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y component of point B

", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "A[1]/A[0]^2", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "(A[1]-b)/A[0]", "description": "

slope

", "templateType": "anything", "can_override": false}, "area": {"name": "area", "group": "Unnamed group", "definition": "- k x^3/3 + m x^2/2 + b x", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "A[0]", "description": "", "templateType": "anything", "can_override": false}, "Qy": {"name": "Qy", "group": "Unnamed group", "definition": "- k x^4/4 + m x^3/3 + b x^2/2", "description": "", "templateType": "anything", "can_override": false}, "xbar": {"name": "xbar", "group": "Unnamed group", "definition": "Qy/area", "description": "", "templateType": "anything", "can_override": false}, "Qx": {"name": "Qx", "group": "Unnamed group", "definition": "(-k^2 x^5/5 + m^2 x^3/3 + m b x^2 + b^2 x)/2", "description": "", "templateType": "anything", "can_override": false}, "ybar": {"name": "ybar", "group": "Unnamed group", "definition": "Qx/area", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "m <> 0 // don't want the line to be horizontal", "maxRuns": 100}, "ungrouped_variables": ["A", "b", "k", "m", "x"], "variable_groups": [{"name": "Unnamed group", "variables": ["area", "Qy", "xbar", "Qx", "ybar"]}], "functions": {"g": {"parameters": [["x", "number"]], "type": "number", "language": "jme", "definition": " m x + by"}, "applet": {"parameters": [], "type": "ggbapplet", "language": "javascript", "definition": "//{geogebra_applet('cpf7gyej',['A': A, 'by': b, \"C\": [visible: false, label_style: 3]])}\n// Create the worksheet. \n// This function returns an object with a container `element` and a `promise` resolving to a GeoGebra applet.\nvar params = {\n material_id: 'cpf7gyej'\n}\n\nvar result = Numbas.extensions.geogebra.createGeogebraApplet(params);\n\n// Once the applet has loaded, run some commands to manipulate the worksheet.\nresult.promise.then(function(d) {\n var app = d.app;\n question.applet = d;\n \n //initialize the dimensions and forces\n \n function setGGBPoint(g_name, n_name = g_name) {\n // moves point in GGB to location of Numbas Vector Variable\n // g_name = geogebra point, n_name = numbas vector\n \n //var pt = question.scope.evaluate(n_name).value\n var pt = scope.getVariable(n_name).value\n app.setFixed(g_name,false,false);\n app.setCoords(g_name, pt[0], pt[1]);\n app.setFixed(g_name,true,true);\n }\n \n function setGGBNumber(gname, nname=gname) {\n // Sets number in GGB to a Numbas Variable\n var n = question.scope.evaluate(nname).value;\n app.setValue(gname,n);\n }\n \n setGGBPoint(\"A\");\n setGGBNumber(\"by\", \"b\");\n app.setVisible(\"C\",false);// centroid\n\n \n});\n\n// This function returns the result of `createGeogebraApplet` as an object \n// with the JME data type 'ggbapplet', which can be substituted into the question's content.\nreturn new Numbas.jme.types.ggbapplet(result);\n"}}, "preamble": {"js": "question.signals.on('adviceDisplayed',function() {\n try{\n var app = question.applet.app;\n app.setVisible(\"C\", true,false);\n }\n catch(err){} \n \n})\n\n\n\n", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "centroid", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

$\\bar{x} = $ [[0]] units

$\\bar{y} = $ [[1]] units

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A value with units marked right if within an adjustable % error of the correct value. Marked close if within a wider margin of error.

Modify the unit portion of the student's answer by

1. replacing \"ohms\" with \"ohm\" case insensitive

2. replacing '-' with ' '

3. replacing '°' with ' deg'

to allow answers like 10 ft-lb and 30°

This fixes the student answer for two common errors.

If student_units are wrong - replace with correct units

If student_scalar has the wrong sign - replace with right sign

If student makes both errors, only one gets fixed.

Only marked close if the student actually has the right sign.