Altitude Cylinder

Let Us Learn About Altitude Cylinder

Introduction to altitude cylinder:

A cylinder is one of the curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity. Its cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder

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Introduction to Weight of the cylinder:

Cylinder is the three dimensional figure. Weight of the cylinder is same as the volume of the cylinder. The formula to find the weight of the cylinder is pi * radius² * height. Height is the total height of the cylinder and radius is the radius of the circular face which is at the bottom and top of the cylinder.

circumference of a cylinder

Let Us Learn About circumference of a cylinder

Introduction for circumference of a cylinder:

A cylinder is a 3-D geometry with two circular surfaces and one curved surface. Let us know how the surface area of a cylinder or circumference is determined. Cylinder has height and radius. The cylinder has two bases, the base has radius r.

Formula for finding circumference of a cylinder.

Circumference of cylinder = 2 x π x r(r + h)

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Introduction to Cylinder volume:

A Cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder.

The surface area and the volume of a cylinder have been known since deep antiquity. In this article of cylinder volume, finding volume of the cylinder is explained.

Formula for Finding Volume of Cylinder:

The diagrammatic representation of a cylinder is shown below:

Formula for finding volume of a cylinder:

Volume of a cylinder = r² h

where, r ----> radius

h----->height

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Cylinder

Let Us Learn About Cylinder

A cylinder is one of the most basic curvilinear geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

In differential geometry, a cylinder is defined more broadly as any ruled surface spanned by a one-parameter family of parallel lines. A cylinder whose cross section is an ellipse, parabola, or hyperbola is called an elliptic cylinder, parabolic cylinder, or hyperbolic cylinder. A prism is a cylinder whose cross-section is a polygon.

A cylinder is one of the most curvilinear basic geometric shapes:It has two faces, zero vertices, and zero edges. The surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since deep antiquity.

Solid line measuring jars, circular pillars, circular pencils, Circular pipes, road rollers and gas cylinders are said to have a cylindrical shape.

There is no edge for the cylinder.

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Pascal’s triangle

Let Us Learn about Pascal's triangle.

The coefficients of the expansions are arranged in an array. This array is
called Pascal’s triangle.

One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher).

Patterns Within the Triangle

Binomial Coefficient

Let Us Understand What Is Binomial Coefficient.

the binomial coefficient is the coefficient of the x ^k term in the polynomial expansion of the binomial power (1 + x) ⁿ.

A binomial coefficient equals the number of combination of r items that can be selected from a set of n items. It also represents an entry in Pascal's triangle. These numbers are called binomial coefficients because they are coefficients in the binomial theorem.

Example:

Binomial Theorem

Let Us Learn What Is Binomial.


The sum of two monomial is called Binomial.

Binomial Theorem for Positive Integral Indices

Let us have a look at the following identities done earlier:

(a+ b)0 = 1 a + b ≠ 0
(a+ b)1 = a + b
(a+ b)2 = a2 + 2ab + b2
(a+ b)3 = a3 + 3a2b + 3ab2 + b3
(a+ b)4 = (a + b)3 (a + b) = a4 + 4a3b + 6a2b2 + 4ab3 + b4

In these expansions, we observe that

(i) The total number of terms in the expansion is one more than the index. For
example, in the expansion of (a + b)2 , number of terms is 3 whereas the index of
(a + b)2 is 2.

(ii) Powers of the first quantity ‘a’ go on decreasing by 1 whereas the powers of the
second quantity ‘b’ increase by 1, in the successive terms.

(iii) In each term of the expansion, the sum of the indices of a and b is the same and
is equal to the index of a + b.

Examples Using Binomial Theorem

Factorial Notation

Let Us Learn About Factorial Notation.


Question:What is Factorial Notation?
Answer: Let n be a positive integer. The continued product of first n natural numbers is called factorial n and is denoted as n

The notation n! represents the product of first n natural
numbers, i.e., the product 1 × 2 × 3 × . . . × (n – 1) × n is denoted as n!. We read this
symbol as ‘n factorial’. Thus, 1 × 2 × 3 × 4 . . . × (n – 1) × n = n !
1 = 1 !
1 × 2 = 2 !
1× 2 × 3 = 3 !
1 × 2 × 3 × 4 = 4 ! and so on.

We define 0 ! = 1
We can write 5 ! = 5 × 4 ! = 5 × 4 × 3 ! = 5 × 4 × 3 × 2 !
= 5 × 4 × 3 × 2 × 1!

Clearly, for a natural number n
n ! = n (n – 1) !
= n (n – 1) (n – 2) ! [provided (n ≥ 2)]
= n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)]
and so on.