The function $g(x,y)$ is a step function taking the following values on each of the rectangles shown below.
\n{domain()}
\nCalculate the integral
\n\\[ \\iint_D g(x,y) dx dy \\]
\nof $g$ over the whole rectangle $D=[0,{\\var{MX()}}]\\times[0,{\\var{N[y]}}]$.
", "advice": "To calculate the integral of the first rectangle $[0, \\var{M[1]}]\\times[0, \\var{N[1]}]$ we just need to find its area and multiply by the value of $g$ inside it, because $g$ is constant.
\n\\[ I_{1,1} = \\int_0^{ \\var{N[1]}}\\int_0^{ \\var{M[1]}} g(x,y)\\, \\mathrm{d}x\\,\\mathrm{d}y = \\var{M[1]} \\cdot \\var{N[1]} \\cdot \\var{Q[1][1]} = \\var{M[1]*N[1]*Q[1][1]} \\]
\nTo integrate over the whole grid we integrate over every other rectangle and add all the results together.
\n\\begin{align}
I =& I_{1,1} &&+ I_{1,2} &&+... \\\\
& \\int_0^{\\var{N[1]}}\\int_0^{\\var{M[1]}} g(x,y)\\, \\mathrm{d}x\\,\\mathrm{d}y &&+ \\int_0^{\\var{N[1]}}\\int_{\\var{M[1]}}^{\\var{M[2]}} g(x,y)\\, \\mathrm{d}x\\,\\mathrm{d}y &&+ ... \\\\
=& \\var{M[1]} \\cdot \\var{N[1]} \\cdot \\var{Q[1][1]} &&+ ( \\var{M[2]}-\\var{M[1]}) \\cdot \\var{N[1]} \\cdot \\var{Q[1][2]} &&+ ... \\\\
=& \\var{M[1]*N[1]*Q[1][1]} &&+ \\simplify{{(M[2]-M[1])*N[1]*Q[1][2]}} &&+ ... \\\\
=& {\\var{ans()}}
\\end{align}
x
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"}, "extensions": ["jsxgraph"], "tags": [], "functions": {"ans": {"language": "javascript", "definition": "var X = Numbas.jme.unwrapValue(scope.variables.x);\nvar Y = Numbas.jme.unwrapValue(scope.variables.y);\nvar Q = Numbas.jme.unwrapValue(scope.variables.q);\nvar M = Numbas.jme.unwrapValue(scope.variables.m);\nvar N = Numbas.jme.unwrapValue(scope.variables.n);\n\nans = 0;\nfor (i = 1; i < X+1; i++) {\n for (j = 1; j < Y+1; j++) {\n ans = (M[i]-M[i-1])*(N[j]-N[j-1])*Q[j][i] + ans\n }\n}\n\nreturn ans;", "parameters": [], "type": "number"}, "NY": {"language": "javascript", "definition": "var Y = Numbas.jme.unwrapValue(scope.variables.y);\nvar N = Numbas.jme.unwrapValue(scope.variables.n);\n\nNY = N[Y];\n\nreturn NY;", "parameters": [], "type": "number"}, "MX": {"language": "javascript", "definition": "var X = Numbas.jme.unwrapValue(scope.variables.x);\nvar M = Numbas.jme.unwrapValue(scope.variables.m);\n\nMX = M[X];\n\nreturn MX;", "parameters": [], "type": "number"}, "domain": {"language": "javascript", "definition": "var X = Numbas.jme.unwrapValue(scope.variables.x);\nvar Y = Numbas.jme.unwrapValue(scope.variables.y);\nvar M = Numbas.jme.unwrapValue(scope.variables.m);\nvar N = Numbas.jme.unwrapValue(scope.variables.n);\n\nM[X+1]=0; M[X+2]=0;\nN[Y+1]=0; N[Y+2]=0;\n\nvar makeboard = Numbas.extensions.jsxgraph.makeBoard(\n '250px','250px', \n {boundingBox:[-3/2,N[Y]+1/2,M[X]+3/2,-3/4], \n axis:true, \n grid:true\n } ); // define how the graph is displayed\n\nvar board = makeboard.board; // generate the graph\n\nvar column1 = board.create('curve',\n [function(y){return M[1];},\n function(y){return y;},\n 0, N[Y]]);\nvar column2 = board.create('curve',\n [function(y){return M[2];},\n function(y){return y;},\n 0, N[Y]]);\nvar column3 = board.create('curve',\n [function(y){return M[3];},\n function(y){return y;},\n 0, N[Y]]);\nvar column4 = board.create('curve',\n [function(y){return M[X];},\n function(y){return y;},\n 0, N[Y]]);\n\nvar row1 = board.create('curve',\n [function(y){return y;},\n function(y){return N[1];},\n 0, M[X]]);\nvar row2 = board.create('curve',\n [function(y){return y;},\n function(y){return N[2];},\n 0, M[X]]);\nvar row3 = board.create('curve',\n [function(y){return y;},\n function(y){return N[3];},\n 0, M[X]]);\nvar row4 = board.create('curve',\n [function(y){return y;},\n function(y){return N[Y];},\n 0, M[X]]);\n\nvar Q = Numbas.jme.unwrapValue(scope.variables.q);\nfor (i = 1; i < X+1; i++) {\n for (j = 1; j < Y+1; j++) {\n p = Q[j][i] // random integer in [-9,9]\n var points = board.create('point', [-2,0],\n {label:{fontSize:15,\n offset:[(125/(M[X]+3))*(4+ M[i-1]+M[i] + 0.10*(M[i-1]-M[i])),\n (125/(N[Y]+5/4))*(N[j-1]+N[j])]}, name:p, fixed:true});\n }\n}\n\nreturn makeboard; // update the board", "parameters": [], "type": "html"}}, "type": "question", "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}]}]}], "contributors": [{"name": "Nicholas Barker", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1915/"}]}