We are asked to evaluate the integral of the absolute value function. To handle the absolute value, we first find the points where the expression inside the absolute value changes sign. This happens when the quadratic expression equals zero.
Step 1: Find the roots of .
We can solve for using the quadratic formula:
where , , and . Thus,
So the roots are:
Step 2: Determine the sign of on different intervals.
We now examine the sign of in the intervals , , and : - For , the quadratic expression is positive because both factors and are negative, making their product positive.
- For , the quadratic expression is negative because is negative, and is positive.
- For , the quadratic expression is positive because both factors and are positive. Thus, we have:
Step 3: Split the integral at and .
We now split the integral as follows:
For , is negative, so:
Thus, the integral from 0 to 1 is:
For , is positive, so:
Thus, the integral from 1 to 2 is:
Step 4: Evaluate the integrals.
We now evaluate both integrals: Thus, the first integral becomes:
Next, we evaluate the second integral:
The second integral gives:
Step 5: Add the results of the two integrals.
Thus, the total value of the integral is: