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Fichas

Multiple Integration

Haris Siddiqui Inglés 200 fichas

Multiple Integration

Fichas del conjunto (200)

  1. 1A definite integral gives usthe area of the graph of a function f(x) between x=a and x=b
  2. 2definite integral =definite integral =
  3. 3the definite integral is thelimit of sum when interval [a,b] is divided by vertical lines into N stripes of the same width and height given by the value of the function inside the integer (called the intergrand) at any point xi within the stripe
  4. 4integrandvalue of the function inside the integral
  5. 5after doing the integral the variable being integrated does whatintegrated disappeares from the result
  6. 6—
  7. 7if the upper limit of the integral is considered as a variable x thenthe result of the integral depends on x (as it is describing the area of f(x') as a function of the upper limit) and therefore relations in a function of xif the upper limit of the integral is considered as a variable x then
  8. 8if we differentiate itwe recover the function inside the integral again
    which is the Fundamental theorem of calculus    if we differentiate it
  9. 9Fundamental theorem of calculusFundamental theorem of calculus
  10. 10integration is the inverse todifferentiation
  11. 11how to calculate integralsby finding antiderivative
    -a function whose derivative is the integrand
  12. 12Double integrals can be done to functions which...depend on at least 2 variables f(x,y)
  13. 13Double integrals representvolume under a graph of the function, limited to a certain area A in the 2D space of the variables (x, y)
  14. 14V =V =
  15. 15the double integral is defined asthe limit of the sum of N small segments of volume whose bottom area is ΔAi and whose heigh is given by the function f(xi,yi) evaluated at any point (xi,yi) inside the area ΔAi
  16. 16the collection all segment areas ΔAi for i=1 to N coversthe entire area in the (x,y) plane
  17. 17—
  18. 18for single integrals the division of the interval [a,b] into differential segments dx canonly be done in one way
  19. 19for double integrals the division of the area A into...can be donethe division of the area A into differential segments dA can be done in many ways
  20. 20integration in rectangular coordinatesusing a rectangular grid of infinitesimal squares of sides dx and dy such that
    dA=dxdy
  21. 21the integral is a limit ofdouble sum of cuboids with base being a tiny square of area ΔA=ΔxΔy and height f(xi,yi), where the double sum uses the indices i and j to label the rectangles across the x and y directions
  22. 22—
  23. 23double integral can be calculated bysimply considering one definite integral after another and this may be done in any order
  24. 24Fubini's TheoremFubini's Theorem
  25. 25what is considered in Fubni's theoremwhat is considered in Fubni's theorem
  26. 26the inner integralthe inner integral
  27. 27representsit is the area of a cut plane of the original functionrepresents
  28. 28from here how to get the desired volumewe integrate g(y)dy, the volume of a differential slide in the interval ? ∈ [?, ?], which results in the desired volumefrom here how to get the desired volume
  29. 29Visual outline of Fubini’s theorem:Visual outline of Fubini’s theorem:
  30. 30—
  31. 31often the integrand can be written as which allows us...product of 2 functions of a single variable each, then we can take out of the integral which does not depend on the integrand variable as if it was a constant
    doing this we can separate the double integral into the product of two integral
  32. 32—
  33. 33the mathematical description of double integrals isdefined as a volume under a graph
  34. 34the single definite integral being defined asan area under a graph
  35. 35the single integral can be interpreted asthe sum of some quantity over a line
  36. 36total mass of a rodtotal mass of a rod
  37. 37total length of a line segmenttotal length of a line segment
  38. 38total distance moved during a time interval, knowing the speedtotal distance moved during a time interval, knowing the speed
  39. 39total displacement(vector) moved during a time interval knowing the velocity(vector)total displacement(vector) moved during a time interval knowing the velocity(vector)
  40. 40total force(vector) acting on a rod (as a function of a "force density" f(x)[N/m])total force(vector) acting on a rod (as a function of a "force density" f(x)[N/m])
  41. 41total electric field created by a linear density of chargetotal electric field created by a linear density of charge
  42. 42average of f(x) over an interval ? ∈ [?, ?] (also weighted average with weight w(x))average of f(x) over an interval ? ∈ [?, ?] (also weighted average with weight w(x))
  43. 43the double integral can be interpreted as theaddition of some quantity over an area
  44. 44Total mass of a planar object covering an area ? (with varying surface density ?(?, ?)):Total mass of a planar object covering an area ? (with varying surface density ?(?, ?)):
  45. 45Total surface area of an area ? (trivial for rectangular areas but useful otherwise!)Total surface area of an area ? (trivial for rectangular areas but useful otherwise!)
  46. 46Total force acting on a wall covering an area ? experiencing a non-uniform pressure ?(?, ?)Total force acting on a wall covering an area ? experiencing a non-uniform pressure ?(?, ?)
  47. 47Total volume above sea level of an island with topography height h(?, ?) with h = 0 being sea level:Total volume above sea level of an island with topography height h(?, ?) with h = 0 being sea level:
  48. 48Total electric field created by a planar object covering an area ? with surface density of charge:Total electric field created by a planar object covering an area ? with surface density of charge:
  49. 49Total price of a plot of land ? whose price per square meter is variable ?(?, ?)Total price of a plot of land ? whose price per square meter is variable ?(?, ?)
  50. 50Average temperature of a surface with non-uniform temperature distribution ?(?, ?):Average temperature of a surface with non-uniform temperature distribution ?(?, ?):
  51. 51Average of any quantity ?(?, ?) over a surface ? (also weighted average with weight ?(?, ?)):Average of any quantity ?(?, ?) over a surface ? (also weighted average with weight ?(?, ?)):
  52. 52Geometric centre and centre of mass (average position weighted by the density) of a planar material:Geometric centre and centre of mass (average position weighted by the density) of a planar material:
  53. 53Double integral were justified byproblem bases on areas and plane 2D objects
  54. 54triple integrals appear in problems related tovolumes of 3D objects
  55. 55triple integrals are defined as I =a limit of an integral sumtriple integrals are defined as I =
  56. 56—
  57. 57similar to the double integral, the triple integral can be interpreted asthe addition of some quantity over a volume
  58. 58Total mass of an object occupying a volume ? (with varying density ?(?, ?, ?)):Total mass of an object occupying a volume ? (with varying density ?(?, ?, ?)):
  59. 59Total volume of a region ΩTotal volume of a region Ω
  60. 60Total electric field created by a 3D object covering a volume ? with varying charge density ?:Total electric field created by a 3D object covering a volume ? with varying charge density ?:
  61. 61Average of any quantity ?(?, ?, ?) over a volume ? (also weighted average with weight ?(?, ?, ?)):Average of any quantity ?(?, ?, ?) over a volume ? (also weighted average with weight ?(?, ?, ?)):
  62. 62Centre of mass (average position weighted by the density) of a 3D object:Centre of mass (average position weighted by the density) of a 3D object:
  63. 63—
  64. 64in a rectangular area the double integral was performed byintegrating alone one direction first with constant limits on integration, and then integrating this function along the other remaining direction
  65. 65—
  66. 66when the region is not rectangularwe have to change the limits of integration to be a function
  67. 67when the region is not rectangularwhen the region is not rectangularwhen the region is not rectangular
  68. 68the integration removes but...the variable of integration, but also adds back any variable which appears in the limits of integration
  69. 69if we first integrate along x when the region is not rectangularwe want the result to be a function of y only with no x, ready for the outer integration, s the limits cannot involve x
  70. 70if we first integrate along xy when the region is not rectangularthen the limits must be functions of x and not involve y
  71. 71the order of integration isan arbitrary choice
  72. 72—
  73. 73—
  74. 74we can define the limits of each successive integration asfunction which depend only on the variables whose integration we have not yet done
  75. 75—
  76. 76—
  77. 77—
  78. 78—
  79. 79in multiple integration we maychange variables (x,y,...) into variables (u,v,...).
  80. 80—
  81. 81how to simplify calculation of a double integralby introducing new variables u and v ai connection equationshow to simplify calculation of a double integral
  82. 82GEOMETRICAL INTERPRETATION OF CHANGE OF VARIABLES IN 2D meansin the definition of the integral instead of chopping the integration area into rectangles bounded by lines of constant x and constant y, we chop it by using curves of consent u and constant v
  83. 83GEOMETRICAL INTERPRETATION OF CHANGE OF VARIABLES IN 2D exampleFor example, we can change the integration to coordinates that are rotated, or to polar coordinates:GEOMETRICAL INTERPRETATION OF CHANGE OF VARIABLES IN 2D example
  84. 84when doing a change of variables we need towhen doing a change of variables we need to
  85. 85Calculate the area of a circle by doing a double integral (with an integrand of 1).Calculate the area of a circle by doing a double integral (with an integrand of 1).
  86. 86when doing a change of variables we need to change the integrandchange the integrand to the new variableswhen doing a change of variables we need to change the integrand
  87. 87when doing a change of variables we need to change the limitschange the limits to make sure they define the same integration region in the new coordinates. The limits may look very different in the (u,v) coordinates, usually simpler if the coordinates are chosen wisely, and may even allow the calc of an integral which was not possible in the previous coordinates
  88. 88when doing a change of variables we need to change the element of areachange the element of area dA = dxdy into some function dA=j(u,v) du dv that account for the area of the new chopping methodwhen doing a change of variables we need to change the element of area
  89. 89how do we know what to write for d? when changing variablesoption 1: find dA by geometrical intuition
    option 2: find dA using the Jacobian
  90. 90how do we know what to write for d? when changing variables option 1how do we know what to write for d? when changing variables option 1
  91. 91how do we know what to write for d? when changing variables option 2how do we know what to write for d? when changing variables option 2
  92. 92The general rule for a change of variables in double integration is therefore:The general rule for a change of variables in double integration is therefore:
  93. 93Calculate the volume of a hemisphere using a double integral in polar coordinates.Calculate the volume of a hemisphere using a double integral in polar coordinates.
  94. 94Calculate the total mass of a disk of radius ? whose density increases with the radius as ?(?) = 1+?2.Calculate the total mass of a disk of radius ? whose density increases with the radius as ?(?) = 1+?2.
  95. 95Change of variables in triple integrals arethe same as in double integrals
  96. 96when having the variable sin triple integrals the Jacobean isnow a 3x3 matrix
  97. 97—
  98. 98—
  99. 99The volume element for commonly used coordinate systems -rectangular -cylindrical -sphericalThe volume element for commonly used coordinate systems -rectangular -cylindrical -spherical
  100. 10010) Calculate the volume of a sphere using a volume integral in spherical coordinates10) Calculate the volume of a sphere using a volume integral in spherical coordinates
  101. 101—
  102. 102—
  103. 103—
  104. 104—
  105. 105—
  106. 106—
  107. 107—
  108. 108—
  109. 109Calculate the volume of a sphere of radius ? using a triple integral (in rectangular coordinates)Calculate the volume of a sphere of radius ? using a triple integral (in rectangular coordinates)
  110. 110Calculate the volume of a cone of radius ? and height h using triple integration.Calculate the volume of a cone of radius ? and height h using triple integration.
  111. 111Calculate the electrostatic potential at (?,?,?)=(0,0,h) for a uniformly charged disk of radius ? with charge density ?. The disk lies in the ?? plane and is centred at the origin. (Remember that the electrostatic potential Φ for a charge ? at a distance ? is given by Φ = ??/?).Calculate the electrostatic potential at (?,?,?)=(0,0,h) for a uniformly charged disk of radius ? with charge density ?. The disk lies in...
  112. 112—
  113. 113—
  114. 114—
  115. 115—
  116. 116example of integration over a curved surfacecalcugitng the avg temp of the surface of the sun or just the surface area of sphere
  117. 117a surface in 3d space can be specified via aa surface in 3d space can be specified via a
  118. 118we must require that each unique value of (?, ?) results inesults in a single position (?, ?, ?). On the contrary, we can allow a same (?, ?, ?) to be represented by several values of (?, ?) (i.e. surfaces may cut across themselves).
  119. 119Examples of surface parametrizations:Examples of surface parametrizations:
  120. 120if we set u =u0, the resulting function r(u0,v) definesa curve in the surface
  121. 121r(u,v0) definesanother curve in the surface
  122. 122the tangent vectors to these curves at the location r(u,v) are given bythe tangent vectors to these curves at the location r(u,v) are given by
  123. 123N = curves in the surfaceN = τu x τv
  124. 124at every location in the surface (u,v)the tangent vectors are always tangent to the surface
  125. 125the two tangent vectors and the jacobins matrixthey form the two columns of the Jaocbina matrix
  126. 126when is the surface smoothand the Katter 2 are non-parallel everywhere inside the domaindefines for u and vwhen is the surface smooth
  127. 127the normal vector to the surface is given bythe cross product of the two tangent vectorsthe normal vector to the surface is given by
  128. 128the normal vector to the surface in terms Jacobean determinantsthe normal vector to the surface in terms Jacobean determinants
  129. 129the normal vector to the surface in terms Jacobean determinants in unit vector formthe normal vector to the surface in terms Jacobean determinants in unit vector form
  130. 130the normal vector to the surface is expressed viathe Jacobins which from its components
  131. 131the arguments in the numerators of the Jacobians together with he unit vectors formthe cyclic sequence x->y->z->x->y
  132. 132whey we equateas transposing a matrix does not change the determinantwhey we equate
  133. 133Calculate the normal vector ? to the outside surface of a cylinder.Calculate the normal vector ? to the outside surface of a cylinder.
  134. 134Calculate the normal vector ? to the outside surface of a sphere.Calculate the normal vector ? to the outside surface of a sphere.
  135. 135how are we supposed to do an integration over a surface or region that is curved?The answer is that we don’t need to! We can map the curved surface into a planar region, by simply using the plane of the parameters (?, ?), and we can carry on with the double integral as usual in (?, ?), as long as the surface area element d? is properly accounted for (d? is the surface element which the little square d? d? in parameter space (?, ?) maps to in the real surface in (?, ?, ?) space).
  136. 136dS=dS = g(u,v)du dvdS=
  137. 137—
  138. 138how to find the surface areahow to find the surface area
  139. 139how do we find the surface element d?Like in change of variables, we have both an intuitive way and a formal recipe to find the surface element d?.
  140. 140options to finding the surface element d?-option 1: find dS by geometrical intuition
    -option2: FIND d? WITH THE NORM OF THE NORMAL VECTOR ? (GENERAL RECIPE)
  141. 141finding the surface element d? find dS by geometrical intuitionfinding the surface element d? find dS by geometrical intuition
  142. 142calculate the area of the outer curved surface of a cylinder of radius R and height H by doing a surface integralcalculate the area of the outer curved surface of a cylinder of radius R and height H by doing a surface integral
  143. 143Calculate the surface area of a sphere of radius ?.Calculate the surface area of a sphere of radius ?.
  144. 144finding the surface element d? find dS with the norm of the normal vector Nfinding the surface element d? find dS with the norm of the normal vector N
  145. 145d? =d? =
  146. 146A double integral of an integrand ?(?, ?) over a surface parametrized by (?, ?) is given by:A double integral of an integrand ?(?, ?) over a surface parametrized by (?, ?) is given by:
  147. 147‖?‖ =‖?‖ =
  148. 148Find d? for the curved surface of a cylinder using the general recipeFind d? for the curved surface of a cylinder using the general recipe
  149. 149Find d? for the surface of a sphere using the general recipeFind d? for the surface of a sphere using the general recipe
  150. 150a surface is given as functionz = f(x,y)
  151. 151r(u,v)=r(u,v)=
  152. 152pt1pt1pt1
  153. 153pt2pt2pt2
  154. 154we can also integrate some scalar or vector field on the surfacewe can also integrate some scalar or vector field on the surface
  155. 155Calculate the area of a surface:Calculate the area of a surface:
  156. 156Calculate the mass of a surface with surface density which varies across the surface ? = ?(?, ?):Calculate the mass of a surface with surface density which varies across the surface ? = ?(?, ?):
  157. 157Average of some magnitude across a surface (e.g. average temperature on Earth ?(?, ?))Average of some magnitude across a surface (e.g. average temperature on Earth ?(?, ?))
  158. 158Center of mass of a surface:Center of mass of a surface:
  159. 159STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL-Define parametrization of the surface ?
    -Obtain the surface differential
    -Evaluate the integrand at the surface (as a function of the parameters)
    -Perform the double integral in the (?, ?) plane [use appropriate limits of a planar double integral such that the parameters cover the whole surface S]
  160. 160STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL -Define parametrization of the surface ?STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL -Define parametrization of the surface ?
  161. 161STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL Obtain the surface differentialSTEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL Obtain the surface differential
  162. 162STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL Evaluate the integrand at the surface (as a function of the parameters)STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL Evaluate the integrand at the surface (as a function of the parameters)
  163. 163STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL Perform the double integral in the (?, ?) plane [use appropriate limits of a planar double integral such that the parameters cover the whole surface S]STEP-BY-STEP INSTRUCTIONS FOR A SURFACE INTEGRAL Perform the double integral in the (?, ?) plane [use appropriate limits of a planar doub...
  164. 164—
  165. 165integrate a vector over a surface, the result beinganother vectorintegrate a vector over a surface, the result being
  166. 166—
  167. 167: be careful when integrating vectors in cylindrical or spherical BASIS, because,as you know, the unit vectors in those bases depend on position, and therefore cannot be taken outside the integrals as constants. So, you cannot integrate the components in cylindrical or spherical coordinates separately. When integrating such vectors, it is convenient to convert them to rectangular basis first, so that ?̂, ?̂, ?̂ being constant can then be taken outside of the integral (i.e. integrate each component separately).
  168. 168—
  169. 169we define the flux of a vector field F(x,y,z) over a surface S aswe define the flux of a vector field F(x,y,z) over a surface S as
  170. 170VECTOR surface differential denoteddS
  171. 171?̂ isa unit vector normal to the surface
  172. 172(? ⋅ ?̂) is givinggiving us a scalar function Fn(u,v) which is the component of the vector field F in the direction ?̂ perpendicular to the surface
  173. 173?̂ =?̂ =
  174. 174dS =dS =
  175. 175N = which iswhich is a vector normal to the surfaceN = which is
  176. 176dS =dS =
  177. 177d? =d? =
  178. 178the flux measureshow much f a field F flows across a surface S per unit time
  179. 179the flux integral representsthe total flow of the quantity through a surface S per unit time
  180. 180the total flux of the quantity through a given surface per unit time isthe total flux of the quantity through a given surface per unit time is
  181. 181STEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL-Define paramerization of the surface s
    -obtain the vector surface differential dS
    -evaluate the integrand at the surface (F⋅d?)
    -Perform the double integral
  182. 182STEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL Define paramerization of the surface sSTEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL Define paramerization of the surface s
  183. 183STEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL obtain the vector surface differential dSSTEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL obtain the vector surface differential dS
  184. 184STEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL evaluate the integrand at the surface (F⋅d?)STEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL evaluate the integrand at the surface (F⋅d?)
  185. 185STEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL Perform the double integralSTEP-BY-STEP INSTRUCTIONS FOR A FLUX INTEGRAL Perform the double integral
  186. 186—
  187. 187—
  188. 188—
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  198. 198pt1pt1pt1
  199. 199pt2pt2pt2
  200. 200do last question in 4.2do last question in 4.2

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Fichas sobre Multiple Integration, creado por Haris Siddiqui el 15/04/2022.
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