### 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.

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.