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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)

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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).

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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

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

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

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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

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Higher Examples: Integration Outcome 2

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

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

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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

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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

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

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

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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.

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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

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

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

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

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Higher Integration Outcome 2 Example :

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

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Calculus Revision Back Next Quit Integrate Integrate term by term

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Calculus Revision Back Next Quit Evaluate Straight line form

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Calculus Revision Back Next Quit Evaluate Straight line form

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Calculus Revision Back Next Quit Integrate Straight line form

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Calculus Revision Back Next Quit Integrate Straight line form

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Calculus Revision Back Next Quit Integrate Straight line form

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Calculus Revision Back Next Quit Integrate Split into separate fractions

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Calculus Revision Back Next Quit Integrate Straight line form

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Calculus Revision Back Next Quit Find p, given

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

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

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

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

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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

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

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Higher Further examples of integration Exam Standard Integration Outcome 2

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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

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Higher Examples: Area under a Curve Outcome 2

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

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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

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

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Higher Example: Area under a Curve Outcome 2

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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

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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

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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

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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

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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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