{plotgraph(4,xmin,xmax,-10,10,a,b,c)}
\nThis is the graph of the function $f(x)=\\simplify{{a}x^3+{b}x^2+{c}x}$.
\n\nWhat is the area bounded by the graph and the $x$-axis (i.e. the total area of the shaded regions)? [[0]]
\nHint: you need to determine the $x$-intercepts before you can do any integration.
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\n\n\n\nAs this is a particularly tricky question, here is some detailed advice:
\n1) Remember that first, you should find an approximation for the answer.
\n\n\n2) To find the minimum and maximum $x$-coordinates of the regions, you need to solve to $f(x)=0$. See Maths 1 for background on this. One of the solutions is 0 and the other roots are $\\var{xmax}$ and $\\var{xmin}$.
\nI label these numbers $a$ and $b$ (so I don't have to keep writing them).
\n\n\n3) I then set up the appropriate integral(s) to calculate the area:
\nArea $ = \\displaystyle \\int_{b}^0 \\simplify{{a}x^3+{b}x^2+{c}x} \\, dx - \\int^{a}_0 \\simplify{{a}x^3+{b}x^2+{c}x} \\, dx$
\n\n\n4) After doing the integration and plugging-in the numbers, you should get $\\var{area_total2}$, which is $\\var{area_total}$, to 3.s.f.
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