(a) What is the equation of the line joining $(0,0)$ to $(\\var{a},\\var{b})$?
\n$y=\\;\\;$[[0]] Input all numbers in your answer as integers or fractions, not as decimals.
\n(b) Evaluate the line integral for $I$:
\n$I=\\;\\;$[[1]] Input your answer as an integer or a fraction, not as a decimal
", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"answer": "{b}/{a}*x", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showPreview": true, "notallowed": {"showStrings": false, "message": "Input all numbers in your answer as integers or fractions, not as decimals.
", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "unitTests": [], "scripts": {}, "answerSimplification": "std", "expectedVariableNames": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 2, "showFeedbackIcon": true}, {"answer": "{a^2+b^2}/{2}", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "variableReplacementStrategy": "originalfirst", "showPreview": true, "notallowed": {"showStrings": false, "message": "Input all numbers in your answer as integers or fractions, not as decimals.
", "strings": ["."], "partialCredit": 0}, "checkingType": "absdiff", "checkVariableNames": false, "vsetRange": [0, 1], "type": "jme", "vsetRangePoints": 5, "unitTests": [], "scripts": {}, "answerSimplification": "std", "expectedVariableNames": [], "checkingAccuracy": 0.001, "showCorrectAnswer": true, "variableReplacements": [], "marks": 6, "showFeedbackIcon": true}], "type": "gapfill", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "Consider the line integral:
\\[I=\\int_{\\Gamma} \\left( \\left(\\simplify[std]{x+y} \\right)\\;dx+\\left(\\simplify[std]{y-x}\\right)\\;dy\\right)\\]
where $\\Gamma$ is the path given by the straight line from $(0,0)$ to $(\\var{a},\\var{b})$
", "tags": ["Calculus", "calculus", "checked2015", "integration", "line integral", "path", "straight line", "Straight Line", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find $\\displaystyle \\int_{\\Gamma} \\left(x+y \\right)\\;dx+\\left(y-x\\right)\\;dy,\\;\\Gamma$ is the line from $(0,0)$ to $(a,b)$.
\n"}, "type": "question", "extensions": [], "advice": "For a line passing through points $(x_1,y_1)$ and $(x_2,y_2)$, the line equation is $\\displaystyle{y-y_1 = m(x-x_1)}$, where the gradient $\\displaystyle{m=\\frac{y_2-y_1}{x_2-x_1}}$. Hence the equation of our line is:
\n\\[\\simplify[std]{y=({b}/{a})*x}\\]
\nSince $\\displaystyle{\\simplify[std]{y=({b}/{a})*x}}$ we have that $\\displaystyle{\\simplify[std]{dy=({b}/{a})*dx}}$. Using these relations we can write the integrand of the line integral in terms of just $x$ or just $y$. We will use $x$. The integrand then becomes:
\\[\\begin{eqnarray*} (x+y)\\;dx+(y-x)\\;dy&=&\\simplify[std]{(x+({b}/{a})*x)dx+(({b}/{a})*x-x)*({b}/{a})*dx}\\\\ &=&\\simplify{{a^2+b^2}/{a^2}}x\\;dx \\end{eqnarray*} \\]
In $x$, our line integral exists over the range $0 \\leq x\\leq\\var{a}$, and so we write our line integral as:
\\[\\begin{eqnarray*} I&=&\\int_\\Gamma\\simplify{{a^2+b^2}/{a^2}}x\\;dx =\\int_0^{\\var{a}}\\simplify{{a^2+b^2}/{a^2}}x\\;dx \\\\ &=&\\simplify{{a^2+b^2}/{a^2}}\\left[\\frac{x^2}{2}\\right]_0^{\\var{a}}\\;dx =\\simplify{{a^2+b^2}/{a^2}}\\times \\frac{\\var{a^2}}{2}=\\simplify[std]{{a^2+b^2}/2} \\end{eqnarray*} \\]