Bài giảng Tích phân (bằng Tiếng Anh)

Higher Unit 2www.mathsrevision.comWhat is IntegrationThe Process of IntegrationArea between to curves.Outcome 2Area under a curveWorking backwards to find function.Area under a curve above and below x-axisIntegrationwe 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).Differentiationmultiply by powerdecrease power by 1Integrationincrease power by 1divide by new powerWhere does this + C come from?IntegrationOutcome 2Integrating is the opposite of differentiating, so: integrateBut:differentiatedifferentiateintegrateIntegrating 6x.......which function do we get back to?IntegrationOutcome 2Solution:When you integrate a function remember to add theConstant of Integration+ CIntegrationOutcome 2means “integrate 6x with respect to x”means “integrate f(x) with respect to x”NotationThis notation was “invented” byGottfried Wilhelm von Leibniz òIntegrationOutcome 2Examples:IntegrationOutcome 2IntegrationOutcome 2Just like differentiation, we must arrange the function as a series of powers of x before we integrate.Name : Integration techniquesArea under curve=Area under curve=IntegrationIntegrationReal Application of IntegrationFind area between the function and the x-axis between x = 0 and x = 5A = ½ bh = ½x5x5 = 12.5Real Application of IntegrationFind area between the function and the x-axis between x = 0 and x = 4A = ½ bh = ½x4x4 = 8A = lb = 4 x 4 = 16AT = 8 + 16 = 24Real Application of IntegrationFind area between the function and the x-axis between x = 0 and x = 2Real Application of IntegrationFind area between the function and the x-axis between x = -3 and x = 3?Houston we have a problem !We need to do separate integrations for above and below the x-axis.Real Application of IntegrationAreas under the x-axis ALWAYS give negative valuesBy convention we simply take the positive value since we cannot get a negative area.Integrate the function g(x) = x(x - 4) between x = 0 to x = 5Real Application of IntegrationWe need to sketch the function and find the roots before we can integrateWe need to do separate integrations for above and below the x-axis.Real Application of IntegrationSince under x-axistake positive valueReal Application of IntegrationFind upper and lower limits.Area between Two Functionsthen integratetop curve – bottom curve.Find upper and lower limits.Area between Two Functionsthen integratetop curve – bottom curve.Take out common factorArea between Two FunctionsTo get the function f(x) from the derivative f’(x) we do the opposite, i.e. we integrate. Hence:IntegrationOutcome 2IntegrationOutcome 2Example :Calculus Revision	BackNextQuitIntegrateIntegrate term by termsimplifyCalculus Revision	BackNextQuitIntegrateIntegrate term by termCalculus Revision	BackNextQuitEvaluateStraight line formCalculus Revision	BackNextQuitEvaluateStraight line formCalculus Revision	BackNextQuitIntegrateStraight line formCalculus Revision	BackNextQuitIntegrateStraight line formCalculus Revision	BackNextQuitIntegrateStraight line formCalculus Revision	BackNextQuitIntegrateSplit into separate fractionsCalculus Revision	BackNextQuitIntegrateStraight line formCalculus Revision	BackNextQuitFind p, givenCalculus Revision	BackNextQuitIntegrateMultiply out bracketsIntegrate term by termsimplifyCalculus Revision	BackNextQuitIntegrateStandard Integral(from Chain Rule)Calculus Revision	BackNextQuitIntegrateSplit intoseparate fractionsMultiply out bracketsCalculus Revision	BackNextQuitEvaluateCannot use standard integralSo multiply outCalculus Revision	BackNextQuitThe graph of passes through the point (1, 2). express y in terms of x. IfsimplifyUse the pointEvaluate cCalculus Revision	BackNextQuitA curve for which passes through the point (–1, 2).Express y in terms of x. Use the pointFurther examples of integrationExam Standard IntegrationOutcome 2The 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 CurveOutcome 2Examples:Area under a CurveOutcome 2Conventionally, the lower limit of a definite integral is always less then its upper limit.Area under a CurveOutcome 2abcdy=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 CurveOutcome 2The Area Between Two CurvesTo find the area between two curves we evaluate:Area under a CurveOutcome 2Example:Area under a CurveOutcome 2Complicated 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 CurveOutcome 291The 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 = t The area of a wing (W ) is given by:Area under a CurveThe area of a rectangle is given by:The area of the complete shaded area is given by:The cargo volume is:Area under a CurveOutcome 2Exam Type QuestionsOutcome 2At this stage in the course we can only do Polynomial integration questions.In Unit 3 we will tackle trigonometry integration