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M04.6 · The fundamental theorem of calculus

A free, illustrated module of 4 lessons and one interactive in M04 Calculus I: Single Variable: The area function; Part one: the slope of the area is the height; Part two: integrals from antiderivatives; Net change and the round trip.

M04.6 L1 · Core 4 lessons · about 100 min Live · in review

Part of M04 Calculus I: Single Variable in the Mathematics track.

Slopes and areas look like two separate subjects, yet each undoes the other. These four lessons turn the area under a curve up to a moving end into a function, prove that its slope is the height of the curve, use that to compute any integral in one line from an antiderivative, and read the result as a rule about rates and totals: integrate a velocity for a displacement, a density for a probability, the slope of a loss for its drop, with NumPy and SciPy on sampled data.

What you need first. Limits, the squeeze theorem and continuity (M04.1), the derivative and its rules (M04.2, M04.3), the mean value theorem (M04.4.1) and the extreme value theorem (M04.4.3), and the definite integral as a limit of sums with its properties and the trapezoid rule (M04.5).

Lessons

1. The area function M04.6.1 25 min

   Fix the left end of an integral and let the right end move, and the area becomes a function of x. Find it from rectangles and triangles, build it as a running total with numpy.cumsum, and notice that its slope is always the height of the curve.

2. Part one: the slope of the area is the height M04.6.2 25 min

   Why the slope of an area function is the height of the curve: its difference quotient is the average height of a thin strip, squeezed between the strip’s lowest and highest values. So every continuous function has an antiderivative, even the bell curve.

3. Part two: integrals from antiderivatives M04.6.3 25 min

   Any two antiderivatives of a function differ by a constant, so the integral from a to b is F(b) - F(a) for any antiderivative F. Read the derivative table backwards, and the running area 1/3 and the trip’s 32/3 metres each take one line.

4. Net change and the round trip M04.6.4 25 min

   Integrating a rate of change gives the total change: velocity gives displacement, a density gives a probability, a slope gives a change in loss. Differentiating and integrating undo each other up to a constant, exactly and on sampled data with NumPy and SciPy.

The interactive

M04.6-X1 · Area accumulator. Drag the upper limit x along the graph of a function and watch the shaded area sweep out while a second panel traces the area function A(x). A thin strip at x shows that A grows at the rate f(x), and falls where f is negative. It lives in Part one: the slope of the area is the height.