MATHEMATICS NOTES FORM FIVE:INTEGRATION

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INTEGRATION

Integration :Is the reverse process of differentiation, i.e. the process of finding the expression for y in terms of x when given the gradient function.

The symbol for integration is E:\..\..\..\thlb\cr\tz\intetgration_files\image001.gif , denote the integrate of a function with respect to x

If E:\..\..\..\thlb\cr\tz\intetgration_files\image002.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image003.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image004.gif

This is the general power of integration it works for all values of n except for n = -1

Example

1.      E:\..\..\..\thlb\cr\tz\intetgration_files\image005.gif

2.       Integrate the following with respect to x
(i)3x2

Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image006.gif

E:\..\..\..\thlb\cr\tz\__i__images__i__\AA9.PNG

            Integration of constant

The result for differentiating c x is c

E:\..\..\..\thlb\cr\tz\intetgration_files\image007.gif

            Properties

(1)  E:\..\..\..\thlb\cr\tz\intetgration_files\image008.gif

(2)  E:\..\..\..\thlb\cr\tz\intetgration_files\image009.gif

Integration by change of variables

If x is replaced by a linear function of x, say of the form ax + b, integration by change of variables will be applied
E.g.     E:\..\..\..\thlb\cr\tz\intetgration_files\image010.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image011.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image012.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image013.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image014.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image015.gif
E:\..\..\..\thlb\cr\tz\__i__images__i__\AAA4.PNG

Considering E:\..\..\..\thlb\cr\tz\intetgration_files\image016.gif in similar way gives the general result

          E:\..\..\..\thlb\cr\tz\intetgration_files\image017.gif

Example

Find the integral of the following

            a) (3x – 8) 6              b) E:\..\..\..\thlb\cr\tz\intetgration_files\image018.gif

Solution (a)

  E:\..\..\..\thlb\cr\tz\intetgration_files\image019.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image020.gif

Solution (b)

E:\..\..\..\thlb\cr\tz\intetgration_files\image021.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image022.gif

            → If E:\..\..\..\thlb\cr\tz\intetgration_files\image023.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image024.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image025.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image026.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image027.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image028.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image029.gif

 

Example

1.   Find  E:\..\..\..\thlb\cr\tz\intetgration_files\image030.gif

            Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image031.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image032.gif

2.   Find E:\..\..\..\thlb\cr\tz\intetgration_files\image033.gif

            Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image034.gif

Integration of exponential function

E:\..\..\..\thlb\cr\tz\intetgration_files\image035.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image036.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image037.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image038.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image039.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image040.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image041.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image042.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image043.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image044.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image045.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image046.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image047.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image048.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image049.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image050.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image051.gif

Example 01

E:\..\..\..\thlb\cr\tz\intetgration_files\image052.gif

            Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image053.gif

Alternative

E:\..\..\..\thlb\cr\tz\intetgration_files\image054.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image055.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image056.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image057.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image058.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image059.gif

Example 02

E:\..\..\..\thlb\cr\tz\intetgration_files\image060.gif

Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image061.gif

Alternative

E:\..\..\..\thlb\cr\tz\intetgration_files\image062.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image063.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image064.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image065.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image066.gif

Integrating fraction

If  E:\..\..\..\thlb\cr\tz\intetgration_files\image067.gif

Differentiating with respect to x gives

            E:\..\..\..\thlb\cr\tz\intetgration_files\image068.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image069.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image070.gif

Example

1.   E:\..\..\..\thlb\cr\tz\intetgration_files\image071.gif ,given that f(x)=x2+1

Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image072.gif

2.      Find E:\..\..\..\thlb\cr\tz\intetgration_files\image073.gif

solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image074.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image075.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image076.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image077.gif

Note: 2x  is the derivative of x2 + 1 in this case substitution is useful

           i.e. let u = x2 + 1

E:\..\..\..\thlb\cr\tz\intetgration_files\image078.gif

   This converts into the form     E:\..\..\..\thlb\cr\tz\intetgration_files\image079.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image080.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image081.gif

E:\..\..\..\thlb\cr\tz\__i__images__i__\gg.PNG

E:\..\..\..\thlb\cr\tz\intetgration_files\image083.gif E:\..\..\..\thlb\cr\tz\intetgration_files\image084.gif

Standard integrals

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image085.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image086.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image087.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image088.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image089.gif →∫sec x tan xdx=sec x+c

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image090.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image091.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image092.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image093.gif

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image094.gif

·        E:\..\..\..\thlb\cr\tz\__i__images__i__\12w.PNG
·       E:\..\..\..\thlb\cr\tz\__i__images__i__\as.PNG

E:\..\..\..\thlb\cr\tz\intetgration_files\image097.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image098.gif

E:\..\..\..\thlb\cr\tz\__i__images__i__\A1213.PNG

·        E:\..\..\..\thlb\cr\tz\intetgration_files\image100.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image101.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image102.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image103.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image104.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image105.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image106.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image107.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image108.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image109.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image110.gif

EXERCISE

Find the integral of the following functions

E:\..\..\..\thlb\cr\tz\intetgration_files\image111.gif i)   E:\..\..\..\thlb\cr\tz\__i__images__i__\AAA5.PNG

ii)   E:\..\..\..\thlb\cr\tz\intetgration_files\image113.gif

iii)  E:\..\..\..\thlb\cr\tz\intetgration_files\image114.gif

iv)  E:\..\..\..\thlb\cr\tz\intetgration_files\image115.gif

Integration by partial fraction

Integration by partial fraction is applied only for proper fraction

E.g.  E:\..\..\..\thlb\cr\tz\intetgration_files\image116.gif

Note that:

 The expression is not in standard integrals

E:\..\..\..\thlb\cr\tz\intetgration_files\image117.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image118.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image119.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image120.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image121.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image122.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image123.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image124.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image125.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image126.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image127.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image128.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image129.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image130.gif

Example 01

E:\..\..\..\thlb\cr\tz\intetgration_files\image131.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image132.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image133.gif

Example 02

E:\..\..\..\thlb\cr\tz\intetgration_files\image134.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image135.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image136.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image137.gif

Improper fraction

If the degree of numerator is equal or greater than of denominator, adjustment must be made

Example

1.      Find  E:\..\..\..\thlb\cr\tz\intetgration_files\image138.gif

Solution

 Both numerator and denominator have the degree of 2

E:\..\..\..\thlb\cr\tz\intetgration_files\image139.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image140.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image141.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image142.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image143.gif

2.      E:\..\..\..\thlb\cr\tz\intetgration_files\image144.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image145.gif

3.      E:\..\..\..\thlb\cr\tz\intetgration_files\image146.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image147.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image148.gif

If the denominator doesn’t factorize, splitting the numerator will work

→ Numerator = A (derivative of denominator) + B

Example

E:\..\..\..\thlb\cr\tz\intetgration_files\image149.gif

Solution

E:\..\..\..\thlb\cr\tz\intetgration_files\image150.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image151.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image152.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image153.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image154.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image155.gif

           Important

It can be shown that

E:\..\..\..\thlb\cr\tz\intetgration_files\image156.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image157.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image158.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image159.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image160.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image161.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image162.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image163.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image164.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image165.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image166.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image167.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image168.gif

EXERCISE

                                I.            E:\..\..\..\thlb\cr\tz\intetgration_files\image169.gif

                             II.            E:\..\..\..\thlb\cr\tz\intetgration_files\image170.gif

                           III.            E:\..\..\..\thlb\cr\tz\intetgration_files\image171.gif

Integrated of the form

E:\..\..\..\thlb\cr\tz\intetgration_files\image172.gif

Note that:

1. If the denominator has two real roots use partial fraction

2. If the denominator has one repeated root use change of variable or recognition

3. If the denominator has no real roots, use completing the square

E.g.

                                I.            E:\..\..\..\thlb\cr\tz\intetgration_files\image173.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image174.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image175.gif

                             II.      E:\..\..\..\thlb\cr\tz\intetgration_files\image176.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image177.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image178.gif

                           III.     E:\..\..\..\thlb\cr\tz\intetgration_files\image179.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image180.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image181.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image182.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image183.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image184.gif

Integral of the form

E:\..\..\..\thlb\cr\tz\intetgration_files\image185.gif

Example
E:\..\..\..\thlb\cr\tz\intetgration_files\image186.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image187.gif

Then hyperbolic function identities is identities is used E:\..\..\..\thlb\cr\tz\intetgration_files\image188.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image189.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image190.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image191.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image192.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image193.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image194.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image165.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image195.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image196.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image197.gif

Note that:

If the quadratic has 1 represented root, it is easier

E.g.

E:\..\..\..\thlb\cr\tz\intetgration_files\image198.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image199.gif

EXERCISE

Find the following

     i.      E:\..\..\..\thlb\cr\tz\intetgration_files\image200.gif

     ii.    E:\..\..\..\thlb\cr\tz\intetgration_files\image201.gif

     iii.   E:\..\..\..\thlb\cr\tz\intetgration_files\image202.gif

     iv.   E:\..\..\..\thlb\cr\tz\intetgration_files\image203.gif

     v.    E:\..\..\..\thlb\cr\tz\intetgration_files\image204.gif

Integration of Trigonometric Expression
Integration of Even power of  E:\..\..\..\thlb\cr\tz\intetgration_files\image205.gif
Note that: for even power of E:\..\..\..\thlb\cr\tz\intetgration_files\image205.gif  use the identity
i) E:\..\..\..\thlb\cr\tz\intetgration_files\image206.gif
ii) E:\..\..\..\thlb\cr\tz\intetgration_files\image207.gif

Example 01

Find  E:\..\..\..\thlb\cr\tz\intetgration_files\image208.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image209.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image210.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image211.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image212.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image213.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image214.gif

Example 02

E:\..\..\..\thlb\cr\tz\intetgration_files\image215.gif
E:\..\..\..\thlb\cr\tz\__i__images__i__\ear.PNG
E:\..\..\..\thlb\cr\tz\intetgration_files\image217.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image218.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image219.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image220.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image221.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image222.gif

            Odd powers of E:\..\..\..\thlb\cr\tz\intetgration_files\image223.gif

For odd powers of E:\..\..\..\thlb\cr\tz\intetgration_files\image224.gif  use identity E:\..\..\..\thlb\cr\tz\intetgration_files\image225.gif

Example

Find    E:\..\..\..\thlb\cr\tz\intetgration_files\image226.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image227.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image228.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image229.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image230.gif
   Any power of tanE:\..\..\..\thlb\cr\tz\intetgration_files\image231.gif
The identity E:\..\..\..\thlb\cr\tz\intetgration_files\image232.gif  is useful as it is the fact that E:\..\..\..\thlb\cr\tz\intetgration_files\image233.gif  It will be understood that;

E:\..\..\..\thlb\cr\tz\intetgration_files\image234.gif

Example:

1.                  Find  E:\..\..\..\thlb\cr\tz\intetgration_files\image235.gif

            Solution:

E:\..\..\..\thlb\cr\tz\intetgration_files\image236.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image237.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image238.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image239.gif

2.      E:\..\..\..\thlb\cr\tz\intetgration_files\image240.gif

solution
E:\..\..\..\thlb\cr\tz\intetgration_files\image241.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image242.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image243.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image244.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image245.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image246.gif

Multiple Angles

To integrate such type of integral, one of the factor formulae will be used

1.  E:\..\..\..\thlb\cr\tz\intetgration_files\image247.gif

2.  E:\..\..\..\thlb\cr\tz\intetgration_files\image248.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image249.gif

3. E:\..\..\..\thlb\cr\tz\intetgration_files\image250.gif

4.  E:\..\..\..\thlb\cr\tz\intetgration_files\image251.gif

Example

1.      Find E:\..\..\..\thlb\cr\tz\intetgration_files\image252.gif

      Solution
E:\..\..\..\thlb\cr\tz\__i__images__i__\as1.PNG
E:\..\..\..\thlb\cr\tz\intetgration_files\image254.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image255.gif

2. E:\..\..\..\thlb\cr\tz\intetgration_files\image256.gif

Solution
E:\..\..\..\thlb\cr\tz\intetgration_files\image257.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image258.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image259.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image260.gif

EXERCISE

Find the integral of the following

1. E:\..\..\..\thlb\cr\tz\intetgration_files\image261.gif

2.   E:\..\..\..\thlb\cr\tz\intetgration_files\image262.gif

3.  E:\..\..\..\thlb\cr\tz\intetgration_files\image263.gif

4. E:\..\..\..\thlb\cr\tz\intetgration_files\image264.gif
Integrated by change of variables E:\..\..\..\thlb\cr\tz\intetgration_files\image265.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image266.gif

   E:\..\..\..\thlb\cr\tz\__i__images__i__\A36.jpg

                 E:\..\..\..\thlb\cr\tz\intetgration_files\image272.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image273.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image274.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image275.gif

1.     E:\..\..\..\thlb\cr\tz\intetgration_files\image276.gif E:\..\..\..\thlb\cr\tz\intetgration_files\image277.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image278.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image279.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image280.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image281.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image282.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image283.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image284.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image285.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image286.gif

Note that

For integrand containing E:\..\..\..\thlb\cr\tz\intetgration_files\image287.gif  and E:\..\..\..\thlb\cr\tz\intetgration_files\image288.gif , or even powers of these, the change of variable  E:\..\..\..\thlb\cr\tz\intetgration_files\image289.gif  can be used.

Example

E:\..\..\..\thlb\cr\tz\intetgration_files\image290.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image291.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image292.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image293.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image294.gif

E:\..\..\..\thlb\cr\tz\intetgration_files\image295.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image296.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image297.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image298.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image299.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image300.gif
E:\..\..\..\thlb\cr\tz\intetgration_files\image301.gif

APPLICATION OF INTEGRATION

To determine the area under the curve

Given A is the area bounded by the curve y=f(x) the x -axis and the line x=0 and x=b where b> a
E:\..\..\..\thlb\cr\tz\__i__images__i__\WW81.png

The area under that curve is given by the define definite integral of f(x) from a to b
E:\..\..\..\thlb\cr\tz\__i__images__i__\AAB.png

          = f (b) – f (a)

Examples

1. Find the area under the curve f(x) =x2+1 from x=0 to x=2

2. Find the area under the curve f(x) = from x=1 to x=2

3. Find the area bounded by the function f(x) =x 2-3, x=0, x=5 and the x- axis

Solution

  1. f(x) =  + 1

      y intercept=1
E:\..\..\..\thlb\cr\tz\__i__images__i__\WW01.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\AZZ.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\AAS.png

  E:\..\..\..\thlb\cr\tz\__i__images__i__\AAX.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\AAK.png

EXERCISE

1.      Find the area between y = 7-x2 and the x- axis from x= -1 to x=2

2.      Find the area between the graph of y=x2 x – 2 and the x- axis from x= -2 to x=3

Solution

1.        y =7-x2

Where y- intercept =7

E:\..\..\..\thlb\cr\tz\__i__images__i__\AAM1.png
= 6.67 + 11.3

=17.97sq units

Volume of the Solids of Revolution

The volume,V of the solid of revolution is obtained by revolving the shaded portion under the curve, y= f(x) from x= a to x =b about the x -axis is given by
E:\..\..\..\thlb\cr\tz\__i__images__i__\AAD.png

Example 1

Find the volume of revolution by the curve y=x2 from x=0 to x=2 given that the rotation is done about the the x- axis

E:\..\..\..\thlb\cr\tz\__i__images__i__\AQ11.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\AWW.png

Exercise

1. Find the volume obtained when each of the regions is rotated about the x – axis.

a) Under  y= x3, from x =0 to x=1
b) Under y2= 4-x, from x=0 to x=2
c)Under y= x2, from x=1 to x=2
d)Under y= √x, from x=1 to x=4

2. Find the volume obtained when each of the region is rotated about the y-axis.
a) Under y= x2, and the y-axis from x=0 to x=2
b) Under y= x3, and the y-axis from y=1 to y=8
c) Under y= √x, and the y-axis from y=1 to y=2

 LENGTH OF A CURVE

                   Consider the curve

E:\..\..\..\thlb\cr\tz\__i__images__i__\Length_of_curev1.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\ccc.png

E:\..\..\..\thlb\cr\tz\__i__images__i__\ggggggggg.png

Example

Find the length of the part of the curves given between the limits:

E:\..\..\..\thlb\cr\tz\__i__images__i__\hhhhhh.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\yyyyyyyyy.png
E:\..\..\..\thlb\cr\tz\__i__images__i__\dddddd.png

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