{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

S5_Unit_2_Outcome_2

# S5_Unit_2_Outcome_2 - Higher Unit 2 Higher Outcome 2 What...

This preview shows pages 1–12. Sign up to view the full content.

www.mathsrevision.com Higher Higher Unit 2 Higher Unit 2 www.mathsrevision.com www.mathsrevision.com What is Integration The Process of Integration ( Type 1 ) Area between to curves ( Type 4 ) Outcome 2 Area under a curve ( Type 2 ) Working backwards to find function ( Type 5 ) Area under a curve above and below x-axis ( Type 3)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
www.mathsrevision.com Higher Outcome 2 Integration 1 ( 1) n n x x dx c n + = + + 4 3 x dx = 5 3 5 x c + 4 1 3 (4 1) x c + + = + we get  You have 1 minute to  come up with the rule. 4 3 x dx = 5 3 5 x c + Integration can be thought of as the opposite of differentiation  (just as subtraction is the opposite of addition).
www.mathsrevision.com Higher Differentiation multiply by power decrease power by 1 Integration increase power by 1 divide by new power n x 1 1 n n x x dx n C + = + + Where does this + C come from? Integration Outcome 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
www.mathsrevision.com Higher Integrating is the opposite of differentiating, so:  ( ) f x ( ) f x integrate 2 2 2 ( ) 3 1 ( ) 3 4 ( ) 3 10 f x x f x x f x x = + = + = - ( ) ( ) ( ) f x g x h x = = = But: differentiate differentiate integrate Integrating  6x ....... which function do we get back to? 6 x Integration Outcome 2
www.mathsrevision.com Higher Solution: When you integrate a function  remember to add the Constant of Integration …………… + C Integration Outcome 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
www.mathsrevision.com Higher 6 x dx means “integrate 6x with respect to x” ( ) f x dx means “integrate f(x) with respect to x” Notation This notation was “invented” by Gottfried Wilhelm von Leibniz  Integration Outcome 2
www.mathsrevision.com Higher Examples: 7 x dx 8 8 x 2 3 2 1 x x dx - + 3 2 3 2 3 2 - + x x x + C 3 2 - + + x x x C Integration Outcome 2

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
www.mathsrevision.com Higher 5 3 1 4 2 x dx x x + + 1 3 5 2 2 ( 4 ) 2 x x x dx - - + + 3 1 4 2 2 3 1 2 2 4 8 x x x C - - + + + - - Integration Outcome 2 Just like differentiation, we  must arrange the function as  a series of powers of x before  we integrate. 3 2 1 4 2 1 2 8 8 3 x C x x - + - + 3 4 1 2 8 8 3 x C x x - + - +
Name : 3 4 x dx 2 2 3 dx x 3 x dx 3 2 1 2 dx x 3 2 3 x dx x + ( 2)(3 1) x x dx - + 2 3 x dx x - 4 1 5 x dx x + Inte g ratio n  te c hniq ue s Are a  unde r c urve = Are a unde r  c urve = Inte g rati on Inte g rati on 4 4 4 x c + 1 2 3 x dx 3 2 3 2 3 x c + 3 2 2 x c + 2 2 3 x dx - 1 2 3( 1) x c - + - 2 3 c x - + 2 3 2 x dx - 1 3 3 2 x c + 3 2 3 x x dx - + 2 2 2 6 2 x x c - + + - 2 2 1 6 x c x - + 1 1 2 4 5 x x dx - + 1 5 2 4 1 5 2 4 5 x x c + + 1 5 2 4 2 4 x x c + + 2 3 5 2 x x dx - - 3 2 3 5 2 3 2 x x x c - - + 1 1 2 2 2 3 x x dx - - 4 x c + 2 3 5 2 2 x x x c - - + 3 1 2 2 3 1 2 2 2 3 x x c - + 3 1 2 2 4 6 3 x x c - +

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Real Application of Integration Find area between the function and  the x-axis  between x = 0 and x = 5 A =  ½  bh =  ½ x 5 x 5 = 12.5 5 0 A x dx = 5 2 0 2 x � � = � � � � 2 2 5 0 25 0 12.5 2 2 2 � � � � = - = - = � � � � � � � �
Real Application of Integration Find area between the function and  the x-axis  between x = 0 and x = 4 A =  ½  bh =  ½ x 4 x

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 49

S5_Unit_2_Outcome_2 - Higher Unit 2 Higher Outcome 2 What...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online