Concept preview

What you will be able to do

1. 01Interpret a definite integral as the signed area between a curve and the horizontal axis over an interval.
2. 02Calculate estimates of a definite integral using left, right and midpoint Riemann sums.
3. 03Determine a definite integral as the limit of Riemann sums as the rectangles become infinitely thin.
4. 04Apply the basic properties of the definite integral, including linearity, additivity and sign.
5. 05Diagnose a flawed area argument, such as confusing signed area with total area.

The definite integral answers a question as old as mathematics: what is the area of a region with a curved boundary. The area of a rectangle or a triangle is elementary, but the area of the region trapped between a curve and the horizontal axis, over some interval of inputs, resists the old formulas because the top is not straight. The definite integral solves this by a strategy that is the twin of the derivative's: approximate the awkward quantity with simple pieces, make the pieces smaller and smaller, and take a limit. Where the derivative used shrinking intervals to define a slope, the integral uses shrinking rectangles to define an area, and the two ideas turn out to be deeply connected, as the next lesson will reveal.