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# Untitled

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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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