// Numbas version: exam_results_page_options {"name": "20. Integration Review", "metadata": {"description": "

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

", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"name": "Family of parabolas", "extensions": ["geogebra"], "custom_part_types": [], "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": ["parabola"], "metadata": {"description": "

Student estimates, then calculates exactly and symbolically the value of $k$ for a parabola $y = k x^2$ which passes through a given point.

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

{applet}

The equation for a parabola with vertex at the origin is given by the general equation

$y=kx^2$.

This equation describes an infinite family of parabolas, each corresponding to a different value of $k$.

", "advice": "

Substituting in the known coordnates of the point into the equation of the parabola for $k$ gives:

$k = \\dfrac{y}{x^2} = \\dfrac{\\var{A[1]}}{\\var{A[0]}^2} = \\var{sigformat(k,3)}$.

The value of $k$ for a parabola passing through the point $(\\var{b}, \\var{h})$ is determined the same way, and the resulting equation for the parabola is

$y= \\dfrac{\\var{h}}{\\var{b}^2}\\; x^2$

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Use the slider to find the approximate value of $k$ ($\\pm 0.1$) required so that the parabola passes through the indicated point. (You may use the arrow keys for finer adjustment of the slider.)

$k_{est} = $ [[0]]

Now substitute the coordinates of the known point into the equation of the parabola and solve for the exact value of $k$ to three significant figures.

$k =$ [[1]]

Determine the equation for a parabola with a vertex at the origin, which passes through a known point located at $(\\var{b}, \\var{h})$.

$y =$ [[2]]

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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.

", "help_url": "", "input_widget": "string", "input_options": {"correctAnswer": "siground(settings['correctAnswer'],4)", "hint": {"static": true, "value": ""}, "allowEmpty": {"static": true, "value": true}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nswitch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n\n\ninterpreted_answer:\nqty(student_scalar, student_units)\n\n\n\ncorrect_quantity:\nsettings[\"correctAnswer\"]\n\n\n\ncorrect_units:\nunits(correct_quantity)\n\n\nallowed_notation_styles:\n[\"plain\",\"en\"]\n\nmatch_student_number:\nmatchnumber(studentAnswer,allowed_notation_styles)\n\nstudent_scalar:\nmatch_student_number[1]\n\nstudent_units:\nreplace_regex('ohms','ohm',\n replace_regex('\u00b0', ' deg',\n replace_regex('-', ' ' ,\n studentAnswer[len(match_student_number[0])..len(studentAnswer)])),\"i\")\n\ngood_units:\ntry(\ncompatible(quantity(1, student_units),correct_units),\nmsg,\nfeedback(msg);false)\n\n\nstudent_quantity:\nswitch(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\n\npercent_error:\ntry(\nscalar(abs((correct_quantity - student_quantity)/correct_quantity))*100 \n,msg,\nif(student_quantity=correct_quantity,0,100))\n \n\nright:\npercent_error <= settings['right']\n\n\nclose:\nright_sign and percent_error <= settings['close']\n\nright_sign:\nsign(student_scalar) = sign(correct_quantity)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "switch( \n right and good_units and right_sign, add_credit(1.0,'Correct.'),\n right and good_units and not right_sign, add_credit(settings['C2'],'Wrong sign.'),\n right and right_sign and not good_units, add_credit(settings['C2'],'Correct value, but wrong or missing units.'),\n close and good_units, add_credit(settings['C1'],'Close.'),\n close and not good_units, add_credit(settings['C3'],'Answer is close, but wrong or missing units.'),\n incorrect('Wrong answer.')\n)\n"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "qty(student_scalar, student_units)\n\n"}, {"name": "correct_quantity", "description": "", "definition": "settings[\"correctAnswer\"]\n\n"}, {"name": "correct_units", "description": "", "definition": "units(correct_quantity)\n"}, {"name": "allowed_notation_styles", "description": "", "definition": "[\"plain\",\"en\"]"}, {"name": "match_student_number", "description": "", "definition": "matchnumber(studentAnswer,allowed_notation_styles)"}, {"name": "student_scalar", "description": "", "definition": "match_student_number[1]"}, {"name": "student_units", "description": "

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": [["question-resources/centroid-integration.ggb", "/srv/numbas/media/question-resources/centroid-integration.ggb"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "tags": ["area", "Area", "definite integral", "Mechanics", "mechanics", "Statics", "statics"], "metadata": {"description": "

Find roots and the area under a parabola

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

{applet()}

Given the function $\\color{red}{y=\\simplify[!noleadingminus,unitfactor]{{A} x^2 + {B} x + {C}}}$, determine the area under the curve from the y-axis to the point where the curve crosses the positive x-axis. The x- and y- coordinates are in [{units}]. You may not use a graphing calculator to solve this problem.

", "advice": "

To find the point where the curve crosses the y-axis, set x = 0 and solve for y.
To find the point where the curve crosses the positive x-axis, set y to zero and solve for x using the quadratic formula. Take the positive root.
To find the area under the curve, divide the shaded area into vertical differential strips where $dA = y\\, dx$ then integrate between the horizontal limits to find the area.

$\\begin{align}A &= \\int dA \\\\&= \\int_0^a y\\,dx \\\\&= \\int_0^a(\\simplify[!noleadingminus,unitfactor]{{A} x^2 + {B} x + {C}})\\, dx\\\\& = \\left[ \\simplify[unitfactor,collectnumbers]{{A} x^3/3 + {B} x^2/2 + {C}x}\\right]_0^\\var{scalar(siground(root1,4))}\\\\&=\\var{siground(area,4)}\\end{align}$

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Determine the points where the curve crosses the axes.

$h$ = [[0]] {qty(C,units)}

$a$ = [[1]] {siground(root1,4)}

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Set up and evaluate the integral equation below to find the area under the curve.

$\\begin{align} A &= \\int{dA}\\\\&=\\int_0^a y\\, dx \\\\ \\vdots\\end{align}$

$A$ = [[0]] {siground(area,4)}

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Student is given two points defined symbolically, and must find the equation of the line they define, then use integration to find an equation for the area under the line, bounded by the x-axis and vertical lines through the two points.

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

{applet}

", "advice": "

The equation of a line passing through two points $A = (\\var{x1}, \\var{y1}) $ and $B = (\\var{x2}, \\var{y2})$ is

$\\dfrac{y-\\var{y1}}{\\simplify[zeroTerm]{x - {x1}}} = \\dfrac{\\var{y2}-\\var{y1}}{\\simplify[zeroTerm]{{x2} - {x1}}}$

which rearranges to:

$y = \\left(\\dfrac{\\var{y2}-\\var{y1}}{\\simplify[zeroTerm]{{x2} - {x1}}}\\right) (\\simplify[zeroTerm]{(x - {x1})}) + \\var{y1}$

Integrate using vertical strips $dA = y dx$ to find the area

$A = \\int dA = \\simplify{defint(y, x, {x1},{x2})}$

$\\phantom{A} = \\int_\\var{x1}^\\var{x2} \\left(\\dfrac{\\var{y2}-\\var{y1}}{\\simplify[zeroTerm]{{x2} - {x1}}}\\right) (\\simplify[zeroTerm]{(x - {x1})}) + \\var{y1}\\; dx$

Simplify by letting $ m = \\frac{\\var{y2}-\\var{y1}}{\\simplify[zeroTerm]{{x2}-{x1}}}$ and performing a $u$-substitiution with

$u = (x - \\var{x1}) $ and $du = dx$

and limits $ u(\\var{x1} )= 0$, and $u(\\var{x2}) = \\var{x2}-\\var{x1}$ to give:

$A = \\int_0^\\simplify{{x2}-{x1}} \\simplify{(m u + {y1})} \\;du$

This function can be easily integrated with respect to $u$, and then expressed in terms of $x$:

$A = m \\dfrac{u^2}{2} + \\var{y1}\\; u \\;\\Big|_0^\\simplify{{x2}-{x1}}$

$\\phantom{A} = \\dfrac{(\\var{y2}-\\var{y1})}{(\\simplify[zeroTerm]{{x2}-{x1}})} \\dfrac{\\simplify{({x2}-{x1})^2}}{2} + \\var{y1}(\\simplify{{x2}-{x1}})$

$\\phantom{A} = \\dfrac{1}{2} (\\var{y2}-\\var{y1})(\\simplify[zeroTerm]{{x2}-{x1}}) + \\var{y1} (\\simplify{{x2}-{x1}})$

The resulting area can be recognized as the sum of the area of a triangle with $\\text{height}= (\\simplify{{y2}-{y1}})$ and $\\text{base}=(\\simplify[zeroTerm]{{x2}-{x1}})$ and the area of a rectangle with $\\text{height}=\\var{y1}$ and the same base.

Alternatively, this further simplifies to the average height of the trapezoid times the base.

$A =\\left( \\dfrac{\\var{y2}+\\var{y1}}{2}\\right) (\\simplify[zeroTerm]{{x2}-{x1}})$

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x = 0 is never selected, but problem will still work. Equations in solution will look poor.

", "templateType": "anything", "can_override": false}, "applet": {"name": "applet", "group": "Ungrouped variables", "definition": "geogebra_file('line_3Hg39hr.ggb',[\n A: A, B: B, \n x1: quote(strings[0]), y1: quote(strings[1]), \n x2: quote(strings[2]), y2: quote(strings[3])])", "description": "", "templateType": "anything", "can_override": false}, "B": {"name": "B", "group": "Ungrouped variables", "definition": "vector(A[0] + random(3,4,5,6), random(1,2,3,4,5))", "description": "", "templateType": "anything", "can_override": false}, "strings": {"name": "strings", "group": "Ungrouped variables", "definition": "random(\n if(A[0]<>0,\n [['a','b','c','d'], ['a','c','b','d'],['x_1','y_1','x_2','y_2']],\n [['0','a','c','b'], ['0','y_1','x_2','y_2'], ['0','h_1','b','h_2']]))", "description": "

picks one of several labeling schemes.

strings needed for ggb labels

", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "expression(strings[0])", "description": "", "templateType": "anything", "can_override": false}, "y1": {"name": "y1", "group": "Ungrouped variables", "definition": "expression(strings[1])", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "expression(strings[2])", "description": "", "templateType": "anything", "can_override": false}, "y2": {"name": "y2", "group": "Ungrouped variables", "definition": "expression(strings[3])", "description": "", "templateType": "anything", "can_override": false}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "substitute([x1: x1, y1: y1, x2: x2, y2: y2], expression(\"(x2-x1)(y2+y1)/2\"))", "description": "

({x2}-{x1})(({y2}-{y1})/2 + {y1})

", "templateType": "anything", "can_override": false}, "AreaNum": {"name": "AreaNum", "group": "Ungrouped variables", "definition": "(A[1]+B[1])/2 (B[0]-A[0])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "a[1]<>B[1] //line is not horizontal \nand abs(A[0]) > 1 // point A not too close to y axis", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "applet", "strings", "x1", "y1", "x2", "y2", "area", "AreaNum"], "variable_groups": [], "functions": {"quote": {"parameters": [["s", "string"]], "type": "string", "language": "jme", "definition": "'\\\"'+s+'\\\"'"}}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "

Use integration to find an expression for the area under the line from $x = \\var{x1}$ to $x = \\var{x2}$.

$A = $ [[0]]

", "gaps": [{"type": "jme", "useCustomName": true, "customName": "Area", "marks": "20", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{area}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "navigatemode": "sequence", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}], "extensions": ["geogebra", "quantities", "weh"], "custom_part_types": [{"source": {"pk": 19, "author": {"name": "William Haynes", "pk": 2530}, "edit_page": "/part_type/19/edit"}, "name": "Engineering Accuracy with units", "short_name": "engineering-answer", "description": "

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.

", "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": [["question-resources/centroid-integration.ggb", "/srv/numbas/media/question-resources/centroid-integration.ggb"], ["question-resources/line_3Hg39hr.ggb", "/srv/numbas/media/question-resources/line_3Hg39hr.ggb"]]}