Final answer:
To evaluate the definite integral of the given function by interpreting it in terms of signed area, we need to find the positive and negative areas between the graph and x-axis for different intervals. Then we sum up these areas to evaluate the definite integral. In this case, the definite integral is equal to 0.
Step-by-step explanation:
To evaluate the definite integral of f(x) = 2x - 4 if 0 ≤ x < 2 and 4 - 2x if 2 ≤ x ≤ 4, we can interpret it in terms of signed area. First, we need to find the area between the graph of f(x) and the x-axis in each interval. For 0 ≤ x < 2, the graph is a line with a positive slope, so the area will be positive. For 2 ≤ x ≤ 4, the graph is a line with a negative slope, so the area will be negative. To evaluate the definite integral, we sum up these areas.
For 0 ≤ x < 2, the area can be found by finding the area of the triangle formed by the line segment from (0, -4) to (2, 0). The base of the triangle is 2 units and the height is 4 units, so the area is (1/2)(2)(4) = 4 square units.
For 2 ≤ x ≤ 4, the area can be found by finding the area of the triangle formed by the line segment from (2, 0) to (4, -4). The base of the triangle is also 2 units and the height is 4 units, so the area is (1/2)(2)(4) = 4 square units. Since the graph has a negative slope, we need to take the negative of this area.
Finally, we can evaluate the definite integral by summing up the areas:
The definite integral of f(x) from 0 to 4 is 4 - 4 = 0.