Full worked solution:
\n$\\begin{array}{rcl}y &=& \\var{a}\\sqrt{-x + \\var{b}}\\\\y^2 &=& \\var{a}^2(-x + \\var{b})\\\\ &=& -\\var{a^2}x + \\var{a^2*b}\\end{array} $
\n$\\begin{array}{rcl}V &=& \\int\\limits_0^\\var{b}\\left(-\\var{a^2}x + \\var{a^2*b}\\right)\\pi\\space\\text{d}x\\\\&=& \\pi\\left[-\\frac{\\var{a^2}x^2}{2} + \\var{a^2*b}x\\right]_0^\\var{b}\\\\&=& \\pi\\left[ (-\\var{(a^2*b^2)/2} + \\var{a^2*b^2}) - (0 + 0)\\right]\\\\&=& \\simplify[fractionNumbers]{{fractionPart}} \\pi\\end{array}$
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\nA mountain is modelled by the solid of revolution of the function $x = \\var{a}\\sqrt{-y + \\var{b}}$ rotated about the $y$-axis over the interval $y = [0,\\var{ylimit}]$. Take the limit of y to 2 decimal places.
\n", "metadata": {"description": "Use integration to find the volume of revolution of a function to estimate the volume of a mountain.
", "licence": "All rights reserved"}, "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "Calculate the volume of the mountain. Give your answer as an exact fraction or decimal.
\nVolume = [[0]] $\\pi$
", "gaps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "allowFractions": true, "minValue": "({a}^2*{ylimit}^2)/2", "correctAnswerFraction": true, "marks": "3", "mustBeReduced": false, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "maxValue": "({a}^2*{ylimit}^2)/2", "showCorrectAnswer": true}], "type": "gapfill", "steps": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "prompt": "\nThe formula for the volume of revolution of a function about the $y$-axis over the interval $[0, a]$ is:
\n$V = \\int\\limits_0^a \\pi x^2 \\space\\text{d}y$
\nWe are given that $x = \\var{a}\\sqrt{-y + \\var{b}}$. Square this to find an expression for $x^2$.
", "type": "jme", "checkvariablenames": false, "answer": "{a}^2*({b}-x)", "showCorrectAnswer": true, "expectedvariablenames": [], "marks": 1, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "showFeedbackIcon": true, "variableReplacements": [], "vsetrange": [0, 1], "showpreview": true}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "prompt": "Substitute this into the volume of revolution equation and integrate to find an expression for the indefinite integral, i.e. before we substitute values of $y$. What expression should fill in the gap $\\left[ (\\text{ _______ }) \\times \\pi\\right]_0^a$?
", "type": "jme", "checkvariablenames": false, "answer": "-({a}^2x^2)/2 + {a}^2*{b}*x", "showCorrectAnswer": true, "expectedvariablenames": [], "marks": 1, "vsetrangepoints": 5, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "showFeedbackIcon": true, "variableReplacements": [], "vsetrange": [0, 1], "showpreview": true}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "We have that the upper limit of integration is $a = \\var{ylimit}$. Substitute $x=\\var{ylimit}$ into the expression in square brackets, then subtract the same expression with $y=0$ substituted instead. Leave your answer in terms of $\\pi$ and you have the final answer for this question.
", "type": "information", "showFeedbackIcon": true, "variableReplacements": [], "showCorrectAnswer": true}], "showFeedbackIcon": true, "variableReplacements": [], "stepsPenalty": "0", "showCorrectAnswer": true}], "rulesets": {}, "type": "question", "contributors": [{"name": "John Hodkinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/921/"}], "resources": []}]}], "contributors": [{"name": "John Hodkinson", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/921/"}]}