A curve dips below the x-axis on my interval — why does the plain integral give too small an area?
A definite integral gives SIGNED area, so any region below the x-axis is counted as negative and cancels part of the positive region above.
A definite integral gives SIGNED area, so any region below the x-axis is counted as negative and cancels part of the positive region above. That's fine for net displacement, but for a true geometric area you don't want cancellation. So split the interval at the points where the curve crosses the axis, integrate each piece, and add the ABSOLUTE values. That way both the above and below parts contribute positively to the total area.
Key point
Split at the roots and sum absolute values to get total (unsigned) area, since integrals give signed area.
Common mistake
reading a signed integral directly as the geometric area.
Memory tip
below-axis parts subtract — split at roots and take |each piece|.
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