Question

Which of the following definite integrals could be used to calculate the total area bounded by the graph of y = x3 - 3x2 + 2x and the x-axis?

a. one half times the integral from 0 to 2 of the quantity x cubed minus 3 times x squared plus 2 times x, dx
b. 2 times the integral from 0 to 1 of the quantity x cubed minus 3 times x squared plus 2 times x, dx
c. the integral from 0 to 1 of the quantity squared of x cubed minus 3 times x squared plus 2 times x, dx
d. the integral from 0 to 2 of the quantity x cubed minus 3 times x squared plus 2 times x, dx

Answer 1

2 times the integral from 0 to 1 of the quantity x cubed minus 3 times x squared plus 2 times x, dx

Explanation:

Given that the graph given is

`y=x^3-3x^2+2x`

This graph has x intercepts as 0,1,2

The region between 0 and 1 is exactly the same as the region between 1 and 2 but other side of x axis.

Hence if we do 0 to 2 integral the value will be shown as 0

Instead we can do integral from 0 to 1 and double it to get both the areas.

Correct answer is

Option b

2 times the integral from 0 to 1 of the quantity x cubed minus 3 times x squared plus 2 times x, dx