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

Question options:

1)
2)
3)
4)

Answer 1

First, to solve this problem we need to find the zeros of f(x)

The equation x^3 - 3x^2 + 2x = 0 has a Greatest Common Factor of x

Therefore we can rewrite this function as x(x^2 - 3x +2) = 0

Next we set each multiple equal to zero to find the zeros

x = 0

x^2 - 3x + 2 = 0

Then factor the quadratic

(x - 2)(x - 1) = 0

now set these multiples equal to zero

x - 2 = 0 x = 2

x - 1 = 0 x = 1

Now we know the zeros are 0, 1, 2

Since this function is bounded by the x-axis and has three zeros the integral will span those three zeros. However this question asks for the total area not simply the integral

If we were to take the integral from 0 to 2 the result would be zero since the curve is symmetric above and below the x axis. With the integral from 0 to 1 being a positive # and the integral from 1 to 2 being a negative # making there sum 0

Therefore to find the total area we will integrate from 0, first zero, to 1, second zero, then multiple by two to find the total area bounded by the curve and the x-axis.

Leaving us with answer choice 2