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  1. Section 7.7:
  2. Approximate Integration
  3. Midpoint rule:
  4. euler(f(x*i)deltax)
  5. integ(f(x)dx~=M_n)from(a->b)
  6. where deltax=(b-a)/n
  7. n=number of rectangles
  8.  
  9. Trapezoidal Rule:
  10. integ(f(x)dx~=T_n)from(a->b)
  11. where deltax=(b-a)/n
  12. and xi=a+ideltax
  13.  
  14. Simpsons rule:
  15. Uses parabolas
  16. integ(f(x)dx)from(a->b)
  17. where
  18. (h/3)[f(a)+4f((a+b)/2)+f(b)]
  19.  
  20. Error:
  21. E_T=integ(f(x)dx)from(a->b) - T_n
  22. # trapezoidal rule^
  23.  
  24. E_M=integ(f(x)dx)from(a->b) - M_n
  25.  
  26. Basically subtract the approx from the actual value
  27.  
  28. Section 8.1:
  29. Arclength
  30. L=lim(n->inf) euler(i=1,n,|P_i-1P_i)
  31. P_i-1P_i=
  32. sqrt((x_i-x_(i-1))^2 + (y_i-y_(i-1))^2)
  33. Arc length formula:
  34. L=integ(sqrt(1+(f'(x))^2))dx from (a->b)
  35.  
  36. Section 8.2:
  37. Area of a Surface of Revolution
  38. L=integ(ds)
  39. C=2pir
  40. so 2pi*f(x) will be for rotating around x axis.
  41.  
  42. ds=sqrt(1+(dy/dx)^2)
  43. or
  44. ds=sqrt(1+(dx/dy)^2)
  45.  
  46. for rotating around x axis
  47. general form:
  48. S=integ(2pi*y)ds from (a->b)
  49. so two options:
  50. S=integ(2pi*y*sqrt(1+(dy/dx)^2))dx from (a->b)
  51. or
  52. S=integ(2pi*y*sqrt(1+(dx/dy)^2))dy from(c->d)
  53.  
  54. for rotating around y axis
  55. general form:
  56. S=integ(2pi*x)ds from (a->b)
  57. radius is x, or g(y)
  58. so two options:
  59. S=integ(2pi*x*sqrt(1+(dy/dx)^2))dx
  60. S=integ(2pi*x*sqrt(1+(dx/dy)^2))dy
  61.  
  62. Section 8.3:
  63. Applications of Physics and Engineering
  64. Finding the centroid
  65.  
  66. m_1(avg(x)-x_1)=m_2(x_2-avg(x))
  67. mass=density*area = P*A
  68. A=integ(f(x)dx)from(a->b)
  69.  
  70. Section 10.1:
  71. Curves defined by parametric functions
  72. You know this.
  73.  
  74. Section 10.2:
  75. Calculus with parametric curves
  76. Chain rule:
  77. (dy/dt)=(dy/dx)*(dx/dt)
  78. If (dx/dt)=/=0 we can find (dy/dx)
  79.  
  80. (dy/dx)=(dy/dt)/(dx/dt)
  81. IMPORTANT:
  82. (d/dx)(y)=((d/dt)(y))/((d/dt)(x))
  83.  
  84. Also, ((d^2)y)/(d(x^2)) which can be found by replacing y.
  85. ((d^2)y)/(d(x^2))=(d/dx)(dy/dx)
  86. =((d/dt)(dy/dx))/((d/dt)(x))
  87. =((d/dt)((dy/dt)/(dx/dt)))/(dx/dt)
  88.  
  89. Arc length of a parametric curve is:
  90. if F' is continuous
  91. then
  92. L=integ(sqrt(1+(f'(x))^2))dx from (a->b)
  93.  
  94. Section 10.3:
  95. Areas and lengths in polar coordinates
  96. A=(1/2)r^2 theta
  97. A=integ((1/2)[f(theta_i *)]^2 deltatheta)
  98.  
  99. example:
  100. r=3sintheta
  101. r=2-sintheta
  102. A=integ((1/2))
  103.  
  104. Section 10.4:
  105.  
  106.  
  107. Section 10.5:
  108. Conic sections:
  109. Parabola with focus (0,p)
  110. directorex = y=-p
  111. X^2 = 4 p Y (opens up(+p)/down(-p))
  112. Y^2 = 4 p X (opens left(-p)/right(+p))
  113.  
  114. Equation of ellipse:
  115. (x^2/a^2)+(y^2/b^2)=1
  116. a>=b>0
  117. verts= +-a,0
  118. foci= +-c,0
  119. c^2 = a^2 - b^2
  120.  
  121. Equation of hyperbola:
  122. (x^2/a^2)-(y^2/b^2)=1 (no y-ints)
  123. vertex:(+-a,0)
  124. foci:(+-c,0)
  125. c^2 = a^2 + b^2
  126. or
  127. (y^2/a^2)-(x^2/b^2)=1 (no x-ints)
  128. vertex:(0,+-a)
  129. foci:(0,+-c)
  130. c^2 = a^2 + b^2
  131.  
  132. Section 10.6:
  133. Conic sections in polar coordinates
  134. Let F be a fixed point (called the focus)
  135. and let l be a fixed line (called the directrix) in a plane
  136. Let e be a fixed positive number (called the eccentricity)
  137. The set of all points P in the plane such that
  138. (|PF|) # Distance between points in plane P and F
  139. /
  140. (|Pl|) # Distance between points in plane P and a fixed line
  141. =
  142. e # eccentricity
  143.  
  144. Circle: e=0
  145. Ellipse: 0<e<1
  146. Parabola: e=1
  147. Hyperbola: e>1
  148.  
  149. Polar equation of the form:
  150. r=(e*d)/(1+-e*cos(theta))
  151. or
  152. r=(e*d)/(1+-e*sin(theta))
  153.  
  154. if +cos(theta) then directorex is x=d
  155. if -cos(theta) then directorex is x=-d
  156.  
  157. if +sin(theta) then directorex is y=d
  158. if -sin(theta) then directorex is y=-d
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