Integration. The Concept of Area (Part I)

Integration. The Concept of Area (Part I)

The Concept of Area is one of the most fundamental concepts in mathematics and science. The word "area" can be defined as the amount of space an object covers, or the amount of space between two objects. The area of a shape or surface is its size. The area of a region can be calculated using trigonometry, which applies to flat surfaces only. The calculation of the area of complicated shapes requires calculus, which involves calculating integrals and applying other concepts from differential calculus.

Circular Disc is a solid figure with a circular base, the edge of which is called its circumference and is often referred to as a rim. A circular disc may also be called a disc or disk. The area of a circular disc depends on its radius and the formula for calculating the area of a circle can be used to calculate the area of a circular disc.

We are going to study the definite integral, which builds on a different idea from the indefinite integral.

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Later we'll see that by the fundamental theorem of calculus there is a close connection between these two types of integrals.

Let's begin with the concept of area. Look at this circular disc of radius r.

We can find the area of a circular disc by using pi. The circumference of a circle is equal to pi times the radius squared. If we know the diameter of a circle, we can use this formula to find its area.

We already know how to find the area of a polygon; this is what we already know. We can try to inscribe a polygon inside a circle, so that all sides of the polygon touch the circle. Another method for finding the area of a circle is by circumscribing it with another polygon.

We would expect the area of this we are going to find to be greater than the area of the inscribed polygon, which is in red, and less than the area of the circumscribed polygon, which is in blue.

The area of the disc is unknown. We should be able to find the area of the inscribed and also circumscribed polygon, however. Suppose the radius of the disc is r, and suppose we divide the inscribed polygon into triangular sectors. Each triangular sector's angle Theta at the centre of the disc equals, where is the number of sides in this triangle.

Thus, using basic trigonometry, one can determine that the area of this sector of the inscribed polygon is equal to r^2 sine theta over two, and cosine theta over two.

Simplifying by the double angle formula, we obtain r^2sin theta over two. Now what is theta? Suppose we use a regular polygon of n sides to circumscribe this disc. Therefore theta is equal to two pi over n.

So this is the area of one piece of the triangle, and same way we can do the circumscribed polygon. We find that this time, the area of this sector of the circumscribing polygon is equal to r^2 tangent theta over two, and again theta equals 2 pi over n.

So, we can compute the area of this sector. Since theta equals two pi over n, this is the area in blue.

Therefore, as we have seen earlier, the area of this circular disc should be greater than or equal to the total area of the inscribed polygon, which is n times this because they are n sectors.

The area of the inscribed polygon is equal to n times the area of the disc. The area of the circumscribed polygon should also be less than this value, since it is n times this value.

As n gets larger and larger, the inscribed polygon will approach the circular disc.

The same is also true for the circumscribing polygon. So let us consider the situation when n tends to infinity of the inscribed polygon.

We can look at the area of this inscribed polygon as n approaches infinity by considering the limit of its expression. We know how to evaluate limits involving mathematical expressions, so this is a straightforward task.

It is easy to see that pi x r^2 equals the area of a circle. In the same way, we can show that the limit of the area of the circumscribed polygon as n approaches infinity is equal to pi times the radius squared.

This is the area of a circumscribed polygon. We can find its area by taking n, the limit as n approaches infinity. This gives us pi x r^2.

Thus, the area of the disc should equal pi x r^2.