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?

Question

asked Oct 22, 2018 50.6k views

1 Answer

Fedo Peralta · Answer 1 · 2018-10-26T09:51:02+0000

Let

. Suppose we define a new function that is

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

is an odd function, since

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

So if

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

and the x-axis is the same from 0 to 1 as it is from 1 to 2. This means

$\int_0^2f(x)\,\mathrm dx=2\int_0^1f(x)\,\mathrm dx$