# Adding continuously

Authored explanation; editorial review pending.

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.

## Worked example

$$
\int_a^b u\frac{dv}{dx}\,dx=[uv]_a^b-\int_a^b v\frac{du}{dx}\,dx
$$

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.

The width and the endpoint term are part of the calculation, not decorations on the integral sign.