The previous exercise had students finding the total area between a curve like x^2 and the x-axis on domains where the range was always positive. To solve these, students could simply blindly set up a definite integral and evaluate.

This article should say not so fast! If we want the total area and the range goes negative sometimes, then we need to integrate in separate pieces.

This open source resource has some examples

For example, perhaps we'd start with the following graph. We'd say let's find the area between the curve and the x-axis. Then we'd go ahead and say alright set up a definite integral. But wait! When we solve this we got 0! Why? Oh no they canceled out.

A different pedagogy and perhaps better would be to say find the area of A, find the area of B, find the area between curve on x = 0 to x = 2. Notice that we have to add absolute value of areas.

Here are some more examples and exercises from that PDF above: