Page 45 - Calculus with Complex Numbers

I n t e g r a I s

4. I Review of real variables

Geometrically the integral represents the area under the graph of f @) between
the limits .x = a, .x = b. The approximating area )(2 flx) dx represents the sum
of the areas of the rectangles height flx) and width dx (Figure 4.1).
W e have the following two theorems.

Theorem 1 (Existence theorem) flx) continuous implies flx) integrable.

Theorem 2 (Fundamental theorem of calculus) lf flx) is continuous for a :jq
.x :jq b, then

where F(x) is any primitive of f (x).

Ffgure 4. /