How do positive and negative regions affect signed area?

Separate two questions: “What is the definite integral?” and “How much geometric area is there altogether?” They do not always have the same answer.

For the definite integral, a region above the horizontal axis counts positively and a region below it counts negatively. If the above-axis region has area A₁ and the below-axis region has area A₂, the net signed area is A₁ − A₂. For total geometric area, count both regions as positive sizes: A₁ + A₂. Equivalently, that total is found by integrating |f(x)|. [S1]

A useful picture description is: trace the graph from left to right, shade each enclosed region, and label each above-axis region “+” and each below-axis region “−.” Keep the labels when finding the integral; remove the negative sign only when finding total geometric area.

Use f(x)=x from x=−1 to x=1. Its graph is a straight line through the origin. From −1 to 0 it lies below the horizontal axis, forming a triangle; from 0 to 1 it lies above, forming another triangle. Each triangle has area 1/2. [S1]

For the signed integral, keep the signs:

Above-axis area − below-axis area = 1/2 − 1/2 = 0.

So ∫₋₁¹ x dx = 0. The positive and negative contributions cancel, but the shaded regions have not disappeared. For total geometric area, add their sizes instead: 1/2 + 1/2 = 1. Thus ∫₋₁¹ |x| dx = 1. [S1]

The zero tells you the net signed contribution is zero; it does not mean there is no geometric area between the graph and the axis.

Split the interval at the places where the graph crosses or meets the horizontal axis, then consider each region separately. For each region, record its geometric size and whether it is above or below the axis. Add the above-axis sizes and subtract the below-axis sizes for the signed integral. Add all the sizes for total geometric area.

For example, if your sketch has above-axis regions with areas 3 and 2, and a below-axis region with area 4, the bookkeeping would be: signed area = 3 + 2 − 4 = 1; total geometric area = 3 + 2 + 4 = 9. These numbers are a made-up illustration of the bookkeeping, not an additional fact about a particular graph.

If a question asks for a definite integral, use the signed result. If it explicitly asks for the total area between the graph and the axis, use the positive sizes. If the wording is unclear, state which quantity you calculated.

You can copy this structure and fill in your own regions:

“On [interval], the graph has above-axis region(s) with total area [A] and below-axis region(s) with total area [B]. Therefore the net signed area, and hence the definite integral, is A − B = [result]. The total geometric area is A + B = [result].”

If one side has no region, enter 0 for its total. If there are several regions on either side, add their geometric sizes within that side before using the formula. In the example f(x)=x on [−1,1], A=1/2 and B=1/2, so the signed result is 0 and the total geometric area is 1. [S1]

The distinction to remember is simple: the integral keeps the below-axis contribution negative; total geometric area counts every region positively.