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Which of the following definite integrals could be used to calculate the total area bounded by the graph of y = x^3 – 3x^2 + 2x and the x-axis?

Which of the following definite integrals could be used to calculate the total area-example-1
User Zubinmehta
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Let
f(x)=x^3-3x^2+2x=x(x-1)(x-2). Suppose we define a new function that is
f transformed so that it is symmetric about the origin. This can be done with


g(x)=f(x+1)=(x+1)(x)(x-1)

Notice that
g is an odd function, since


g(-x)=(-x+1)(-x)(-x-1)=-(x-1)(x)(x+1)

So if
g(x) is symmetric about the origin, this means
f(x) must by symmetric about the point
(1,0). This in turn means that the area between
f and the x-axis is the same from 0 to 1 as it is from 1 to 2. This means


\int_0^2f(x)\,\mathrm dx=2\int_0^1f(x)\,\mathrm dx
User Fedo Peralta
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