M.Nabeel Khan 61Daud Mirza 57Danish Mirza 58Fawad Usman 66Amir Mughal 72M.Arslan 17

-Archimedes is the founder of surface areas and volumes of solids such as the sphere and the cone. His integration method was very modern since he did not have algebra, or the decimal representation of numbers

-Gauss was the first to make graphs of integrals. -Leibniz and Newton discovered calculus and

found that differentiation and integration undo each other

Gauss

• -Integration was used to design PETRONAS

Towers making it stronger• -Many differential equations were used in the

designing of the Sydney Opera House• was one of the first uses of integration• -Finding areas under curved surfaces,• Centers of mass, displacement and • Velocity, and fluid flow.

( ) '( ) ( ) ( ) ( ) '( )

f x g x dx f x g x g x f x dx

udv uv vdu

• The idea is to use the above formula to simplify an integration task.• One wants to find a representation for the function to be integrated in the form udv so that the function vdu is easier to integrate than the original function.

• The rule is proved using the Product Rule for differentiation.

Example 1:

cos x x dxpolynomial factor u x

u dv uv v du

sin cosx x x C

u v v du

sin sin x x x dx

Example 2:

ln x dxlogarithmic factor

u dv uv v du

lnx x x C

1ln x x x dxx

u v v du

This is still a product, so we need to use integration by parts again.

Example 3:2 xx e dx

2 2 x xx e e x dx 2 2 x xx e xe dx

2 2x x xx e xe e dx

2 2 2x x xx e xe e C

Example 4(cont.):

cos xe x dx

cos xe x dx 2 cos sin cosx x xe x dx e x e x

sin coscos 2

x xx e x e xe x dx C

sin sinx xe x x e dx

sin cos cos x x xe x e x e x dx

sin cos cos x x xe x e x x e dx