Formulas

Question 1

Washer method formula vs. cylinder method formula

Answer 1

washer:
pi * int( (outer)^2 - (inner)^2 )dx

cylinder:
2*pi * int( x * f(x) )dx

Question 2

washer and cylinder method WHEN AXIS IS NOT X OR Y AXIS

Answer 2

washer:
pi*(int( outer^2 - inner^2 )) d whatevs

cylinder:

2piint( (x+d) (f(x)-g(x) )
^could adjust EITHER the radius in 2pir OR the functions, but DONT do both

Question 3

integral of sinax

Answer 3

-1/a *cosax + c

Question 4

integral of cosax

Answer 4

1/a*sinax + c

Question 5

what is cos^2 ax =?

Answer 5

1/2 * (1 + cos(2a)x)

Question 6

what is sin^2 ax =?

Answer 6

1/2 * (1-cos(2a)x)

Question 7

curve length formula

Answer 7

inta, bdx

Question 8

Pythagorean trig identities:

Answer 8

Sin^2x + cos^2x = 1
1 + tan^2x = sec^2 x
1 + cot^2x = csc ^2 x

Question 9

work formula

Answer 9

int[starting end position, final position (generally from 0 to height)] ( mass density * acceleration * (total length - y) )

Question 10

what is the key to solving volume work problems with cones?

Answer 10

similar triangles at different heights and the proportions therein

Question 11

30-60-90 triangle side lengths

Answer 11

just like the unit circle!
bottom leg (cos) = 1/2, other leg = rad3/2 (sin), and hypoteneuse = 1!

Question 12

45-45-90 side lengths

Answer 12

like pi/4 on a unit circle, hyp = 1 and rad2/2 for the rest

Question 13

derivative of inverses of these functions (in order)
1. sinx
2. cosx
3. tanx
4. cscx
5. secx
6. cotx

Answer 13

1. 1/ sqrt( 1-x^2 )
2. • 1 / sqrt( 1-x^2 )
3. 1/( 1 + x^2 )
4. • 1/( abs(x) * sqrt( x^2-1 ) )
5. 1/( abs(x) * sqrt( x^2-1 ) )
6. • 1 /( 1 + x^2 )

Question 14

what is cos2x equal to?

Answer 14

1. cos^2(x) - sin^2(x)
2. 1 - 2sin^2(x)
3. 2cos^2(x)-1
4. (1-tan^2(x))/(1+tan^2(x))

Question 15

basic integration formula for e^(ax)

Answer 15

int( e^(ax) ) dx = 1/a * e^(ax) + c

Question 16

integral of sec^2(7x)?

Answer 16

1/7 * tan(7x)

Question 17

tanx reduction formula

Answer 17

int(tan^n x dx) = tan^(n-1)x/(n-1) - int( tan^(n-2) x dx)

Question 18

dividing fraction by whole number vs. dividing whole number by fraction?

Answer 18

fraction/whole number = multiply whole number into denominator

whole number/fraction = move bottom of fraction to top

Question 19

formula for work of chain

Question 20

formula for work of trough/pumping liquid

Question 21

domain of inverse trig functions?

Question 22

derivative of tan?

Answer 22

sec^2 (x)

Question 23

what is sin(2x) equal to?

Answer 23

2sinxcosx

Question 24

domain of tangent?

Answer 24

-pi/2 to pi/2 (due to sin/cos if cos = 0 then undefined)

Question 25

what is tan2x equal to?

Answer 25

2tanx/(1-tan^2(x))

Question 26

how to find integral of secx? (and what it is)

Answer 26

multiply by sectan/sectan ln(abs(sec + tan) )

Question 27

how to choose trig substitution.... (that table with 3 scenarios involving x and a)

Answer 27

a^2 - x^2 x = a*sin(theta) a^2 + x^2 x = a*tan(theta) x^2 - a^2 x = a * sec(theta)

Question 28

integral of b^x

Answer 28

1/(lnb) * b^x

Question 29

hyperbolic equations sinh(x) = ? cosh(x) = ? tanh(x) = ? csch(x) = ? sech(x) = ? coth(x) = ?

Answer 29

sinh(x) = (e^x - e^-x )/2 cosh(x) = (e^x + e^-x)/2 tanh(x) = (e^x - e^-x)/(e^x + e^-x) csch(x) = 2/(e^x - e^-x) sech(x) = 2/(e^x + e^-x) coth(x) = (e^x + e^-x)/(e^x - e^-x)

Question 30

integral of 1/( sqrt(a^2 - x^2) )?

Answer 30

arcsin(x/a) + c, a > 0

Question 31

Simpson's rule formula

Answer 31

delta x = (b-a)/n, where a and b are lower and upper bounds, n is number of intervals S(n) = delta x/3 * [ f(x_0) + 4f(x_1) + 2f(x_2).... + 2f(x_n-2) + 4f(x_n-1) + f(x_n) ] starts counting at indices 0, then goes up to nth value (inclusive) always attached to a 1/3 n MUST BE EVEN

Question 32

trapezoid area

Answer 32

(top + bot)/2 * h

Question 33

integral of tan(ax)

Answer 33

-1/a * ln(cos(ax))

Question 34

formula for geometric series

Answer 34

sigma from k = 1 to n-1 (a*r^k) where a is the first term in the series basically constant to power of variable and k = n-1 if abs(r) < 1 lim is equal to lim of ( a/(1-r) )

Question 35

geometric partial sum formula

Answer 35

S(n) = a*( (1-r^n)/(1-r) ) where n-1 = k (number at top of sigma minus the number at the bottom)

Question 36

when does something diverge, according to the divergence test?

Answer 36

when the limit of the expression inside the sigma is not 0

Question 37

p-series test?

Answer 37

p-series = basically variable to power of constant sum of (1/k^p) converges for p > 1, diverges for p <= 1

Question 38

describe convergence test vs. divergence test vs. limit test vs. squeeze theorem vs. telescoping series test vs. p-series test vs. geometric series test

Question 39

for what value of pe does the series sigma(k = 10, infinity)[ 1/k^p ] converge (initial index = 10)? for what values does it ddiverge?

Answer 39

the p-series converges for p > 1, and diverges for p <=1

Question 40

what does the integral test say

Answer 40

the integral of some f(x) (where ak = f(x) = sigma ak behave the same