The Area Between Two CurvesWhat If … ?When f(x) < 0Another ProblemSolutionGeneral SolutionTry This!Area Between Two CurvesThe Area of a Shark FinAnother PossibilitySlicing the Shark the Other WayPracticeHorizontal SlicesAssignmentsThe Area Between Two CurvesLesson 6.1What If … ?•We want to find the area between f(x) and g(x) ?•Any ideas?When f(x) < 0•Consider taking the definite integral for the function shown below.•The integral gives a negative area (!?)We need to think of this in a different wayabf(x)( )baf x dx�Another Problem•What about the area between the curve and the x-axis for y = x3•What do you get forthe integral?•Since this makes no sense – we need another way to look at it232x dx-�Solution•We can use one of the properties of integrals•We will integrate separately for -2 < x < 0 and 0 < x < 2( ) ( ) ( )b c ba a cf x dx f x dx f x dx 2 0 23 3 32 2 0x dx x dx x dx- -= +� � �We take the absolute value for the interval which would give us a negative area.We take the absolute value for the interval which would give us a negative area.General Solution•When determining the area between a function and the x-axisGraph the function firstNote the zeros of the functionSplit the function into portions where f(x) > 0 and f(x) < 0Where f(x) < 0, take absolute value of the definite integralTry This!•Find the area between the function h(x)=x2 – x – 6 and the x-axisNote that we are not given the limits of integrationWe must determine zeros to find limitsAlso must take absolutevalue of the integral sincespecified interval has f(x) < 0Area Between Two Curves•Consider the region betweenf(x) = x2 – 4 and g(x) = 8 – 2x2•Must graph to determine limits•Now consider function insideintegralHeight of a slice is g(x) – f(x)So the integral is [ ]22( ) ( )g x f x dx--�The Area of a Shark Fin•Consider the region enclosed by•Again, we must split the region into two parts0 < x < 1 and 1 < x < 9( ) 9 9 ( ) 9f x x g x x x axis= - = - -Another Possibility•To calculate the region enclosed by•Calculate the area of both regionsSubtract the small "triangular" region from the larger region( ) 9 9 ( ) 9f x x g x x x axis= - = - -Slicing the Shark the Other Way•We could make these graphs as functions of y•Now each slice isy by (k(y) – j(y))( ) 9 9 ( ) 9f x x g x x x axis= - = - -( )2 21( ) 9 ( ) 99j y x y and k y x y= = - = = -[ ]30( ) ( )k y j y dy-�Practice•Determine the region bounded between the given curves•Find the area of the region26y x y x= = -Horizontal Slices•Given these two equations, determine the area of the region bounded by the two curvesNote they are x in terms of y228x yx y= -=Assignments•Lesson 6.1•Page 361•Exercises 1 – 29
View Full Document
Unlocking...