4. Integration
Area Problem: (Method of Exhaustion).
Formulas for the areas of plane regions with straight-line boundaries (squares,
rectangles, triangles, trapezoids, etc.) were well known in many early
civilizations.
On the other hand, obtaining formulas for regions with curvilinear boundaries (a
circle being the simplest case) caused problems for early mathematicians.
The first real progress on such problems was made by the Greek mathematician,
Archimedes, who obtained the areas of regions bounded by arcs of circles,
parabolas, spirals, and various other curves by ingenious use of a procedure later
known as the method of exhaustion.
That method, when applied to a circle of radius r, consists of inscribing a
succession of regular polygons in the circle and allowing the number of sides n to
increase indefinitely.
As n increases, the polygons tend to “exhaust” the region inside the circle, and
the areas of those polygons become better and better approximations to the exact
area of the circle.
4
7. Integration
THE RECTANGLE METHOD FOR FINDING AREAS:
• To find the area under the curve y = x2 over the interval [0, 1], we
will begin by dividing the interval [0, 1] into n equal sub intervals,
from which it follows that each subinterval has length 1/n; the
endpoints of the subintervals occur at
• The heights of our rectangles will be
• Then the area will be the sum of all
individual rectangles
------------ (1)
7
8. Integration
THE RECTANGLE METHOD FOR FINDING AREAS (Cont):
For n = 4
Table below shows the result of evaluating eq (1)
on a computer for some increasingly large
values of n. These computations suggest
that the exact area is close to1/3
8
9. Integration
AREA vs Integration:
is the area of a trapezoid with parallel sides of lengths 1 and 2x + 3
and with altitude x - (-1) = x + 1. For this area function,
• That is, the derivative of the area function A(x) is
the function whose graph forms the upper boundary
of the region.
• This is called an antidifferentiation
problem because we are trying to find
A(x) by“undoing” a differentiation.
9
17. Integration
• PROPERTIES OF INDEFINITE INTEGRAL.
• These equations must be applied carefully to avoid errors and unnecessary
complexities arising from the constants of integration. For example, if you
were to use (4) to integrate 0x by writing
17
