Graphics Reference
In-Depth Information
10
Integration
The Fundamental Theorem of Calculus
Preliminary reading:
Gardiner ch. III.3, Toeplitz ch. 2.
Concurrent reading:
Bryant ch. 5, Courant and John ch. 2.
Further reading:
Spivak chs 13 and 14.
Areas with curved boundaries
Circumscribed
polygon
Inscribed
polygon
Figure10.1
The idea of integration comes from the need to measure areas. The
measurement of area is the measurement of the quantity of units of
surface needed for an exact covering. From the use of a unit square
comes the area of a rectangle as length times breadth. Since a triangle
can be dissected and reassembled to form a rectangle, the area of a
triangle can be calculated, and since any area bounded by a polygon
can be dissected into triangles, the area of any polygon can also be
calculated. However, when the boundary is curved, the measurement of
area is more diMcult. The areas of inscribed and circumscribed
polygons provide lower and upper bounds on the area (see figure 10.1),
and if the construction of inscribed and circumscribed polygons can be
done progressively in such a way that the areas of inscribed polygons
