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www.mathsrevision.com Higher Higher Unit 2 www.mathsrevision.com What is Integration The Process of Integration ( Type 1 ) Area between to curves ( Type.

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Presentation on theme: "www.mathsrevision.com Higher Higher Unit 2 www.mathsrevision.com What is Integration The Process of Integration ( Type 1 ) Area between to curves ( Type."— Presentation transcript:

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2 Higher Higher Unit 2 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)

3 Higher Outcome 2 Integration we get You have 1 minute to come up with the rule. Integration can be thought of as the opposite of differentiation (just as subtraction is the opposite of addition).

4 Higher Differentiation multiply by power decrease power by 1 Integration increase power by 1 divide by new power Where does this + C come from? Integration Outcome 2

5 Higher Integrating is the opposite of differentiating, so: integrate But: differentiate integrate Integrating 6x … which function do we get back to? Integration Outcome 2

6 Higher Solution: When you integrate a function remember to add the Constant of Integration …………… + C Integration Outcome 2

7 Higher means “integrate 6x with respect to x” means “integrate f(x) with respect to x” Notation This notation was “invented” by Gottfried Wilhelm von Leibniz  Integration Outcome 2

8 Higher Examples: Integration Outcome 2

9 Higher Integration Outcome 2 Just like differentiation, we must arrange the function as a series of powers of x before we integrate.

10 Name : Integration techniques Area under curve = Area under curve = Integration

11 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

12 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 4 = 8 A = lb = 4 x 4 = 16 A T = = 24

13 Real Application of Integration Find area between the function and the x-axis between x = 0 and x = 2

14 Real Application of Integration Find area between the function and the x-axis between x = -3 and x = 3 ? Houston we have a problem !

15 We need to do separate integrations for above and below the x-axis. Real Application of Integration Areas under the x-axis ALWAYS give negative values By convention we simply take the positive value since we cannot get a negative area.

16 Integrate the function g(x) = x(x - 4) between x = 0 to x = 5 Real Application of Integration We need to sketch the function and find the roots before we can integrate

17 We need to do separate integrations for above and below the x-axis. Real Application of Integration Since under x-axis take positive value

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19 Find upper and lower limits. Area between Two Functions then integrate top curve – bottom curve.

20 Find upper and lower limits. then integrate top curve – bottom curve. Take out common factor

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22 Higher To get the function f(x) from the derivative f’(x) we do the opposite, i.e. we integrate. Hence: Integration Outcome 2

23 Higher Integration Outcome 2 Example :

24 Calculus Revision Back Next Quit Integrate Integrate term by term simplif y

25 Calculus Revision Back Next Quit Integrate Integrate term by term

26 Calculus Revision Back Next Quit Evaluate Straight line form

27 Calculus Revision Back Next Quit Evaluate Straight line form

28 Calculus Revision Back Next Quit Integrate Straight line form

29 Calculus Revision Back Next Quit Integrate Straight line form

30 Calculus Revision Back Next Quit Integrate Straight line form

31 Calculus Revision Back Next Quit Integrate Split into separate fractions

32 Calculus Revision Back Next Quit Integrate Straight line form

33 Calculus Revision Back Next Quit Find p, given

34 Calculus Revision Back Next Quit Integrate Multiply out brackets Integrate term by term simplify

35 Calculus Revision Back Next Quit Integrate Standard Integral (from Chain Rule)

36 Calculus Revision Back Next Quit Integrate Split into separate fractions Multiply out brackets

37 Calculus Revision Back Next Quit Evaluate Cannot use standard integral So multiply out

38 Calculus Revision Back Next Quit The graph of passes through the point (1, 2). express y in terms of x. If simplify Use the point Evaluate c

39 Calculus Revision Back Next Quit A curve for which passes through the point (–1, 2). Express y in terms of x. Use the point

40 Higher Further examples of integration Exam Standard Integration Outcome 2

41 Higher The integral of a function can be used to determine the area between the x-axis and the graph of the function. NB:this is a definite integral. It has lower limit a and an upper limit b. Area under a Curve Outcome 2

42 Higher Examples: Area under a Curve Outcome 2

43 Higher Conventionally, the lower limit of a definite integral is always less then its upper limit. Area under a Curve Outcome 2

44 Higher a b cd y=f(x) Very Important Note: When calculating integrals: areas above the x-axis are positive areas below the x-axis are negative When calculating the area between a curve and the x-axis: make a sketch calculate areas above and below the x-axis separately ignore the negative signs and add Area under a Curve Outcome 2

45 Higher The Area Between Two Curves To find the area between two curves we evaluate: Area under a Curve Outcome 2

46 Higher Example: Area under a Curve Outcome 2

47 Higher Complicated Example: The cargo space of a small bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. Find the area of this cross- section and hence find the volume of cargo that this ship can carry. Area under a Curve Outcome 2 9 1

48 Higher The shape is symmetrical about the y-axis. So we calculate the area of one of the light shaded rectangles and one of the dark shaded wings. The area is then double their sum. The rectangle: let its width be s The wing: extends from x = s to x = t The area of a wing (W ) is given by: Area under a Curve

49 Higher The area of a rectangle is given by: The area of the complete shaded area is given by: The cargo volume is: Area under a Curve Outcome 2

50 Higher Exam Type Questions Outcome 2 At this stage in the course we can only do Polynomial integration questions. In Unit 3 we will tackle trigonometry integration

51 Higher Outcome 2 Are you on Target ! Update you log book Make sure you complete and correct ALL of the Integration questions inIntegration the past paper booklet.


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