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4.2. The Fundamental Theorem of Calculus

Let x be an indefinite integral of \(\dot{x}\), and let \(\dot{x}\) be a continuous function (one whose graph is a single connected curve). Then

\[ \int_{a}^{b} \dot{x}(t) dt = x(b)-x(a) . \]

The fundamental theorem is proved on page 154. The idea it expresses is that integration and differentiation are inverse operations. That is, integration undoes differentiation, and differentiation undoes integration.

Example 53

◊ Interpret the definite integral

\[ \int_{1}^{2} \frac{1}{t}dt . \]

graphically; then evaluate it it both symbolically and numerically, and check that the two results are consistent.

hyperbola-area
Figure D. The definite integral \(\int_1^2 (1/t)dt\).

◊ Figure d shows the graphical interpretation. The numerical calculation requires a trivial variation on the program from example 51:

  a := 1;
  b := 2;
  H := 1000;
  dt := (b-a)/H;
  sum := 0;
  t := a;
  While (t<=b) [
    sum := N(sum+(1/t)*dt);
    t := N(t+dt);
  ];
  Echo(sum);

The result is 0.693897243, and increasing H to 10,000 gives 0.6932221811, so we can be fairly confident that the result equals 0.693, to 3 decimal places.

Symbolically, the indefinite integral is x=ln t. Using the fundamental theorem of calculus, the area is ln 2 - ln 1 ≈ 0.693147180559945.

Judging from the graph, it looks plausible that the shaded area is about 0.7.

This is an interesting example, because the natural log blows up to negative infinity as t approaches 0, so it's not possible to add a constant onto the indefinite integral and force it to be equal to 0 at t=0. Nevertheless, the fundamental theorem of calculus still works.