Foundation · Explanatory preview
Adding continuously
An integral adds contributions from narrow intervals; its units include the width of each interval.
Original explanatory text; editorial review pending. This is not a translated source passage.
What operation does an integral describe here?
Approximate the area under a curve by rectangles. Each contributes height times width. Add all contributions, then refine the widths. When those sums approach a limit, that limit is the integral.
For a probability density, a rectangle has units of inverse length times length, leaving a dimensionless probability. For total probability the whole area must be one.
Integration by parts comes from adding the product rule for differentiation over an interval: the integral of u times the derivative of v equals the endpoint product minus the integral of v times the derivative of u.
One worked example
Integration by parts moves a derivative from one factor to another and retains the endpoint term.
A constant density of 0.25 per micrometre over 4 micrometres gives 0.25 times 4 = 1, regardless of how many equal rectangles we use.
A stopping point: The width and the endpoint term are part of the calculation, not decorations on the integral sign.
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