Almost everything will work again if you unplug it for a few minutes, including you. Anne Lamott
Idea Transcript
7/17/97
Final Exam: \Cheat Sheet" 1. 2. 3. 4. 5.
Z
1 dx = ln jxj x ax ax dx = lna
Z Z Z Z
cos x dx = sin x
9.
sec x dx = tanx
10.
13. Cylindrical Shells: V = b a
Z
8.
Z
b a
Z
a
b
csc x cot x dx = , csc x dx = 1 tan, x x +a a a x dx p = sin, a a ,x 1
Z
2
2
1
2
2
Z
b a
Z
d
c
[f(y) , g(y)] dy Z
[f(x)] dx. Around the y-axis: V =
d
2
c
[g(y)] dy 2
2xf(x) dx where 0 a < b
f(x)g0 (x) dx = f(x)g(x)]ba , R
Z
sec x tan x dx = sec x
[f(x) , g(x)] dx. In terms of y: A =
12. Volume around the x-axis: V =
14.
Z
sin x dx = , cos x
11. Area in terms of x: A =
csc x dx = , cot x 2
7.
2
Z
Z
6.
Z
b
f 0 (x)g(x) dx
a m n sin x cos x dx.:
15. How to Evaluate If m or n are odd use sin x = 1 , cos x and cos x = 1 , sin x, respectively and then apply substitution. Otherwise, use sin x = (1,cos 2x) or cos x = (1+cos 2x) or sin x cos x = sin 2x. R 16. How to Evaluate tanm x secn x dx.: If m is odd or n is even, use tan x = sec x , 1 and sec x = 1 + tan x, respectively and then apply substitution. 17. Table of Trigonometric Substitutions Expression Substitution Identity p a , x x = a sin, , 1 , sin = cos p