Slice the rectangles thinner, and the area gets exact
See how approximating the area under a curve with several rectangles (a Riemann sum) works, and check how the part that dips below the x-axis counts as negative area.
A shape curved like a curve can't use the plain rectangle-area formula directly. So Riemann sums approximate the area by slicing the space under the curve into many thin rectangles. The more rectangles you use, the less overhang there is, and taking the limit as the number of rectangles grows without bound, the approximation becomes exactly equal to the true area (the definite integral).
The definite integral has a curious property. When the graph dips below the x-axis, that region's area gets subtracted instead of added. So unlike an ordinary shape's area, the value of a definite integral can even come out negative.
Use the slider to increase the number of rectangles n and watch the area get closer to the exact value, and see with your own eyes how area below the x-axis actually gets subtracted.
Riemann sums ā We want to find the area between f(x)=xĀ² and the x-axis (0ā¤xā¤2). Approximate the exact curved shape by slicing it into n rectangles, and the sum of each rectangle's area gets closer to the true area. Increase n with the slider and watch the sum approach the exact value (8/3).
Number of rectangles n6
ā«āĀ² xĀ² dx = 8/3 ā 2.667
Signed area ā f(x)=x moves between below (negative) and above (positive) the x-axis at x=0. A definite integral calculates the "net area," adding the area above the x-axis and subtracting the area below it. Unlike an ordinary shape's area, the value of a definite integral can even come out negative.