# Differential and Integral Calculus 2021

## 7. Area

Table of Content

### Area in the plane

We consider areas of plane sets bounded by closed curves. In the more general cases, the concept of area becomes theoretically very difficult.

The area of a planar set is defined by reducing to the areas of simpler sets. The area cannot be "calculated", unless we first have a definition of "area" (although this is common practice in school mathematics).

#### Starting point

##### The area of a rectangle

The area of a rectangle is##### Definition: Area of a Parallelogram

The area of a parallelogram is

##### Definition: Area of a triangle

The area of a triangle is (by definition) \[ A=\frac{1}{2}ah. \]

#### Polygon

A (simple) **polygon** is a plane set bounded by a closed curve that
consists of a finite number of line segments without self-intersections.

##### Definition: Area of a polygon

The area of a polygon is defined by dividing it into a
finite number of triangles (called a

##### Theorem.

The sum of the areas of triangles in a triangulation of a polygon is the same for all triangulations.

#### General case

For a plane set \(\color{red} D\) bounded by a closed curve we can construct inner polygons \(\color{blue}P_i\) and outer polygons \(P_o\): \(\color{blue}P_i\color{black} \subset \color{red}D\color{black}\subset P_o\).

**A surprise:** The condition that \(D\) is bounded by a closed curve (without self-intersections)
does not guarantee that it has an area! Reason: The boundary curve can be so "wiggly",
that it has positive "area". The first such example was constucted by [W.F. Osgood, 1903]:

**Wikipedia: Osgood curve**

##### Example

Derive the formula \(A=\pi R^2\) for a circle with radius \(R\) by choosing regular inscrided and circumscribed \(n\)-gons as inner and outer polygons, and let \(n\to\infty\).

The solution is a voluntary exercise, where you need the limit \[\lim_{x\to 0}\frac{\sin x}{x} = 1.\] Hint: Show that the inscribed and circumscribed areas are \[ \pi R^2\frac{\sin (2\pi/n)}{2\pi/n} \ \text{ and }\ \pi R^2\frac{\tan \pi/n}{\pi/n}.\]