Area Under a Curve

If a graph shows speed against time, what does the area under it represent?

It is the distance

Speed times time is distance, and each thin strip of area is a speed times a small time.

Estimating with rectangles

Divide the region into rectangles and add them. Narrower rectangles give a better estimate.

The definite integral

The limit of those sums, as the rectangles become infinitely thin, is the definite integral.

Accumulation, before any antiderivative

The area under a rate curve is the total accumulated. Under a velocity graph it is distance travelled; under a flow rate it is volume. The integral is defined as an accumulation, independently of how it is computed.

Approximating with rectangles

Slice the region into thin rectangles and add their areas. More, thinner rectangles give a better estimate. The exact area is the limit as the width goes to zero, which is where the integral comes from.

Area below the axis counts negative

A definite integral computes signed area, so regions below the axis subtract. For total distance rather than displacement, the negative parts must be handled separately — the integral alone gives displacement.

The notation records the construction

The ∫ is a stretched S for sum, and dx is the vanishing width. The symbol literally reads as a sum of function values times widths, which is exactly what the limit of rectangles is.