The fundamental theorem of calculus

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The fundamental theorem of calculus

Let 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 in Proof of the fundamental theorem of calculus. The idea it expresses is that integration and differentiation are inverse operations. That is, integration undoes differentiation, and differentiation undoes integration.

Example

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 4.4

The definite integral

$\int_{1}^{2}(1/t)dt$

Figure 4.4 shows the graphical interpretation. The numerical calculation requires a trivial variation on the program from Example:

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 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 $x=\textrm{In }t$ . Using the fundamental theorem of calculus, the area is $\textrm{In }2 - \textrm{In }1 \approx 0.693147180559945$ .

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

This is an interesting example, be- cause the natural log blows up to negative infinity as 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.

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The fundamental theorem of calculus

The fundamental theorem of calculus

Example