<\/span><\/h2>\nTo find the definite integral of an absolute value function, you need to break the function into multiple pieces based on the intervals where it changes its behavior. Here are the steps to do it:<\/p>\n
1. Determine the critical points of the absolute value function by setting the expression inside the absolute value equal to zero and solving for the variable.
\n2. Create intervals on the number line based on the critical points found in the previous step.
\n3. Within each interval, eliminate the absolute value by using the sign conditions. Rewrite the function without the absolute value symbol.
\n4. Integrate the absolute value function piecewise within each interval.
\n5. Finally, evaluate the definite integral by subtracting the value of the integral at the lower limit from the integral at the upper limit.<\/p>\n
To illustrate this process, let’s consider the example of finding the definite integral of the absolute value function |x – 2|.<\/p>\n
Step 1: The critical point is found by setting x – 2 equal to zero: x = 2.<\/p>\n
Step 2: We create two intervals: (-\u221e, 2) and (2, \u221e).<\/p>\n
Step 3: Within the first interval (-\u221e, 2), the absolute value becomes -(x – 2). Within the second interval (2, \u221e), the absolute value becomes (x – 2).<\/p>\n
Step 4: Integrate each piece of the function within their respective intervals:
\n\u222b[-(x – 2)] dx for (-\u221e, 2) and \u222b[(x – 2)] dx for (2, \u221e).<\/p>\n
Step 5: Evaluate the definite integral by subtracting the value at the lower limit from the value at the upper limit.<\/p>\n
**The definite integral of the absolute value function |x – 2| from a to b is given by \u222b[-(x – 2)] dx from a to 2 plus \u222b[(x – 2)] dx from 2 to b.**<\/p>\n