Solving a integral problem by using substitution method

Topic:- Integration

Integration is an important concept together with differentiation, forms one of the main operations in calculus help
Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral

is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b.

Let's work out a simple example on this.
Question:-

solve ∫ ( x / √x²-9 )

Answer:-

We do it by substitution method.
 Let u = x2-9
    
    du
    ---- = 2x
    dx
 
   du
or ---- = xdx
   2
substituting these values ,the integral becomes

∫ (du/2) / √u

= 1/2 ∫ (u)-1/2 du

        (u)-1/2 + 1
= 1/2 ---------------
         -1/2 + 1
by equalizing the denominators
   
    -1+2
   ------- = 1/2
      2
So the integral becomes
    
        (u)1/2
= 1/2 ---------------   + c
         1/2

= (x2-9)1/2+C (as u =x2-9)

= √( x2-9 ) +c  is the Answer

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A problem on Integrating Factor

Topic:Integrating Factor

In simple term integrating factor in mathematics, is an function that is chosen to facilitate
the solving of a given equation involving differentials. It is generally used to solve ordinary differential equations, but is also used within multivitamin calculus,in this case often multiplying through by an integrating factor allows an inexact differential to be made into an exact differential (which can then be integrated to give a scalar field).
In simple term integrating factor is denoted as I.F.

Question:
   
           dy
solve     ____ + = ex
           dx

Answer:

   Standard form: dy
                 ____ + p(x) * y = q(x)    Here p(x) = 1
                  dx                            q(x) = ex


 Now Inergrating factor = e√pdx = e√idx = ex

 Now the equation is:

                     y = i
                        ___
                        I.F  √I.F * Q(x) dx + c

Plugging in all the values:

                     y = 1
                        ___
                         ex √(ex * ex) dx + c

                     y = 1
                        ___
                         ex √(ex)2 dx + c

                     y = 1 * e2x
                        ___ ___ + c
                        (ex) 2

                     y = ex
                        ___ + c
                         2        

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a problem on quadrilaterals

Topic:-quadrilateral

In geometry, a quadrilateral is a polygon with four 'sides' or edges and four vertices or corners. Sometimes, the term quadrangle is used, for analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The word quadrilateral is made of the words quad and lateral . Quad means four and lateral means sides. The interior angles of a quadrilateral add up to 360 degrees of arc.

Question:-

In a quadrilateral, the third angle is 2 times the fourth and the second angle is 3 times the fourth. The first angle is given as 120. Find the fourth angle.

Solution:-

Let the fourth angle be 'x'.

Here, the third angle = 2 times the fourth angle = 2x.

The second angle = 3 times the fourth angle = 3x.

We know the sum of the angles of a quadrilateral = 360°.

Therefore x + 2x + 3x + 120 = 360.

ie 6x + 120 = 360.

ie 6x = 360 - 120 = 240.

ie x = 240 / 6 = 40.

Hence the fourth angle = 40°.

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Solving Linear Equations

Topic:- Linear Equation


A mathematical expression that has an equal sign and linear expressions is called a Linear equation.

A number that you don't know,often represented by "x" or "y" is called a variable.

We solving linear equations by clicking and dragging a number to the "other" side of the equal sign. Remember that you are "isolating" the unknown "X" to solve the problem.
(Example is provided below.)

Question:-

Solve 2x-18=15

Answer:-

This math help is a part of linear equations.

2x-18=15

Add 18 to both sides

2x-18+18 = 15 + 18

2x = 15+18

2x = 33


divide by 2 on both sides


2x      33
----- = -------
2        2


  x = 16.5

16.5 is the Answer.

How to find volume of a cylinder

Topic :- Volume of cylinder

A cylinder is one of the most basic curvilinear geometric shape, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder.

Here is one sample problem on how you can find the volume.

Question:-
In a cylinder radius is 3 inches and height is 8 inches ,find the volume of the cylinder.

Solution:-

Formula for cylinder volume= Π r² h


So Π =3.14

r =radius=3

h =height=8

Hence, Volume of the cylinder

= 3.14 * (3)²*8

= 3.14 * 9 * 8

= 226.08


So the volume of the cylinder is
226.08 cubic inches.

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A problem on Simplifying Algebraic Fractions

Fractions is a basic section of math where student get to learn how to do fractions, types of fractions and how to simplify fractions involving algebraic terms or numeric terms as shown in the below problem.


Topic : Simplifying Algebraic Fractions

Below problem is an algebra question which explains how to do fraction.

Problem : Show the steps that you would take to solve the following algebraically




What potential solution did you obtain? Explain why this is not a solution.

Solution :










=> 3 * 2 = 2x - 3x + 9

=> 6 = - x + 9

=> - x + 9 = 6

=> - x = 6 - 9

=> -x = - 3

=> x = 3

This is not a solution because at x = 3, the denominators of the terms at the right hand side and the left hand side become ‘zero’. So, x = 3 is only a potential solution and not a solution.

Using Shell or Disc Method to Find Volume of the Solid

Disc method and Shell(cylinder) method of integration are the two different methods of finding volume of solid of a revolution, using rectangular coordination system the functions are defined in terms of x in the below problem.

Topic : Disc or Cylinder Method of Finding Volume of the Sphere.
Problem : Use the disc or shell method to find the volume of the solid generated by revolving the regions bounded by the graphs of the equations about the specified line y = 8.
y=x³ y=0, x=2
Solution :
Here the solid is rotated along x-axis
y = x³ => x = (y)^1/3

or x = y^1/3

when y = 8, x = (8)^1/3 = (2³)^1/3 = 2^3*1/3 = 2

So a = 0, b = 2

Volume of a Solid by rotating about y = 8 is given by:

V = 2π₀∫²(y)^1/3 . y dy

= 2π₀∫²(y)^4/3 dy

= 2π[y^7/3/(7/3)₀]²

= 2π[(2)^7/3/(7/3)- (0)^7/3/(7/3)]

= 2π[((2⁷)^1/3/(7/3)]

= 2π ³√(128)/(7/3)

= 6.³√(128)π/7

= 4.32π