Presentation on theme: "Higher Unit 2 Outcome 2 What is Integration"— Presentation transcript:
1 Higher Unit 2 Outcome 2 What is Integration The Process of Integration ( Type 1 )Area under a curve ( Type 2 )Area under a curve above and below x-axis( Type 3)Area between to curves ( Type 4 )Working backwards to find function ( Type 5 )
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).we get
3 Where does this + C come from? IntegrationOutcome 2Differentiationmultiply by powerdecrease power by 1divide by new powerincrease power by 1IntegrationWhere does this + C come from?
4 IntegrationOutcome 2Integrating is the opposite of differentiating, so:differentiateintegrateBut:differentiateintegrateIntegrating 6x… which function do we get back to?
5 Integration Constant of Integration……………+ C Outcome 2 Solution: When you integrate a functionremember to add theConstant of Integration……………+ C
6 ò Integration Outcome 2 Notation means “integrate 6x with respect to x”means “integrate f(x) with respect to x”This notation was “invented” byGottfried Wilhelm von Leibnizò
8 Just like differentiation, we must arrange the function as a series of powers of x before we integrate.IntegrationOutcome 2
9 Integration techniques Areaunder curve=IntegrationArea under curve=Name :
10 Real Application of Integration Find area between the function and the x-axisbetween x = 0 and x = 5A = ½ bh = ½x5x5 = 12.5
11 Real Application of Integration Find area between the function and the x-axisbetween x = 0 and x = 4A = ½ bh = ½x4x4 = 8A = lb = 4 x 4 = 16AT = = 24
12 Real Application of Integration Find area between the function and the x-axisbetween x = 0 and x = 2
13 ? Houston we have a problem ! Real Application of Integration Find area between the function and the x-axisbetween x = -3 and x = 3?Houston we have a problem !
14 Real Application of Integration By convention we simply take the positive value since we cannot get a negative area.Areas under the x-axis ALWAYS give negative valuesReal Application of IntegrationWe need to do separate integrations for above and below the x-axis.
15 Real Application of Integration Integrate the function g(x) = x(x - 4) between x = 0 to x = 5We need to sketch the function and find the roots before we can integrate
16 Real Application of Integration We need to do separate integrations for above and below the x-axis.Since under x-axistake positive value
33 Calculus Revision Multiply out brackets Integrate term by term simplifyBackNextQuit
34 Calculus Revision Standard Integral (from Chain Rule) Back Next Quit IntegrateStandard Integral(from Chain Rule)BackNextQuit
35 Calculus Revision Multiply out brackets Split into separate fractions IntegrateMultiply out bracketsSplit intoseparate fractionsBackNextQuit
36 Cannot use standard integral Calculus RevisionEvaluateCannot use standard integralSo multiply outBackNextQuit
37 passes through the point (1, 2). Calculus RevisionThe graph ofpasses through the point (1, 2).Ifexpress y in terms of x.simplifyUse the pointEvaluate cBackNextQuit
38 passes through the point (–1, 2). Calculus RevisionA curve for whichpasses through the point (–1, 2).Express y in terms of x.Use the pointBackNextQuit
39 Further examples of integration Outcome 2Further examples of integrationExam Standard
40 Area under a Curve Outcome 2 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.
42 Area under a Curve Outcome 2 Conventionally, the lower limit of a definite integralis always less then its upper limit.
43 When calculating integrals: Area under a CurveOutcome 2abcdy=f(x)Very Important Note:When calculating integrals:areas above the x-axis are positiveareas below the x-axis are negativeWhen calculating the area between a curve and the x-axis:make a sketchcalculate areas above and below the x-axis separatelyignore the negative signs and add
44 Area under a Curve Outcome 2 The Area Between Two Curves To find the area between two curves we evaluate:
46 Area under a Curve Outcome 2 9 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.9Find the area of this cross-section and hence find the volume of cargo that this ship can carry.1
47 Area under a Curve The rectangle: let its width be s 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 sThe wing: extends from x = s to x = tThe area of a wing (W ) is given by:
48 Area under a Curve Outcome 2 The cargo volume is: The area of a rectangle is given by:The area of the complete shaded area is given by:The cargo volume is:
49 Exam Type Questions At this stage in the course we can only do Outcome 2At this stage in the course we can only doPolynomial integration questions.In Unit 3 we will tackle trigonometry integration
50 Are you on Target ! Update you log book Make sure you complete and correctALL of the Integration questions inthe past paper booklet.