everything u and ur mom need to know so you don't bomb the final, bitch. study tf up
Center of mass- equation of mass
integral f(x)-g(x) dx
Center of mass- My
integral x[f(x)-g(x)] dx
Center of mass- Mx
integral 1/2[f(x)²-g(x)²] dx
Center of mass- x center of mass
My/m
Center of mass- y center of mass
Mx/m
What is M?
Sum of individual moments of x or y axes
Arc length formula
integral sq rt[1+f’(x)²] dx
Surface area x-axis
integral 2pi f(x) [arc length for f’(x)]
Surface area y-axis
integral 2pi f(y) [arc length for f’(y)]
First 3 steps to approaching an integral?
Recognize it, algebra, u-substitution
5 basic approaches
U-substitution, partial fractions, long division, completing square, multiplying by 1
sin2x=
2sinxcosx
cos2x=
cos²x-sin²x, 1-2sin²x, 2cos²x-1
Integration by parts formula
integral udv = uv- (integral vdu)
Integral of lnx
xln - x + C
Power reducing formula cos²x
(1+cos2x)/2
Power reducing formula tan²x
(1-cos2x)/(1+cos2x)
Power reducing formula sin²x
(1-cos2x)/2
What is an improper integral?
Integral where one or both of the bounds is infinity
If the limit of an improper integral exists?
Converges
If the limit of an improper integral doesn’t exist?
Diverges
Trig substitution a²-x²
x = asin(theta)
Trig substitution x²-a²
x = asec(theta)
Trig substitution a²+x²
x = atan(theta)
How to rationalize substitutions?
u = sq rt of an expression, square both sides and find derivative, substitute for x and dx and use long division/partial fractions to solve
Weierstrass substitution dx=?
2/(1+t²)
sinx and cosx trig identity
sin²x + cos²x = 1
secx and tanx trig identity
sec²x - tan²x = 1
cscx and cotx trig identity
csc²x - cot²x = 1
Formula for geometric series
[a(1-r^n)]/(1-r)
Formula for infinite geometric series
a/(1-r)
Geometric series converges:
|r| < 1
Geometric series diverges:
|r| > 1 or = 1
Divergence test diverges:
lim ak is not 0
Divergence test is inconclusive:
lim ak = 0
Integral test converges:
integral ak < infinity
Integral test diverges:
integral ak = infinity
P-series form
1/k^p
P-series converges:
p > 1
P-series diverges:
p < 1 or = 1
Ratio test converges:
lim a(k+1)/ak < 1
Ratio test diverges:
lim a(k+1)/ak > 1
Root test converges:
lim (ak)^1/k < 1
Root test diverges:
lim (ak)^1/k > 1
Root test and ratio test are inconclusive if lim=?
1
Limit comparison test converges:
0 </= lim ak/bk < infinity and bk CONVERGES
Limit comparison test diverges:
lim ak/bk > 0 and bk DIVERGES
Alternating series form
(-1)^k(ak)
Alternating series converges:
lim ak = 0
Alternating series diverges:
lim ak is not 0
Alternating series converges absolutely:
|ak| converges
Alternating series converges conditionally:
|ak| diverges and ak converges
Alternating series diverges absolutely:
ak diverges (by divergence test)
d/dx cotx
-csc²x
d/dx secx
secxtanx
d/dx cscx
-cscxcotx
integral tanx
ln|secx| + C
integral secx
ln|secx + tanx| + C
weierstrass substitution t=?
t=tan1/x