Module 2 Formulas

Question 1

V = ?

Answer 1

V = integral of a to b A(x)dx

Question 2

Disks
y = f(x): ?
x = g(y): ?

Answer 2

y = f(x):
V = integral a to b A(x)dx
V = integral a to b pi[R(x)]^2dx
x = g(y):
V = integral c to d A(y)dy
V = integral c to d pi[R(y)]^2dy

Question 3

Washers
y = f(x): ?
x = g(y): ?

Answer 3

y = f(x):
V = integral a to b A(x)dx
V = integral a to b pi([R(x)]^2-[r(x)]^2)dx
x = g(y):
V = integral c to d A(y)dy
V = integral c to d
pi([R(y)]^2-r(y)]^2)dy

Question 3

Arclength
y = f( x): ?
x = g(y): ?

Answer 3

y = f( x):
L = integral a to b
Squareroot(1+[f(x)]^2)dx L = integral c to d Squareroot(1+(dy/dx)^2)dx x = g(y): L = integral a to b Squareroot(1 +[g(yl]^2)dy
L = integral a to b
Squareroot(1+(dx/dy)^2) dy

Question 4

Work
Constant Force: W = ?
Variable Force: W = ?
Hooke’s Law for Springs: W = ?

Answer 4

Constant Force:
W = Fd
Variable Force:
W = integral a to b F(x)dx
Hooke’s Law for Springs:
F =kx
W = integral a to b kxdx
Remember Units (SI and British)
Force: Newton/Pound
Distance: Meter/Feet/Inches
Work: Newton-meter/Joule/Ft-lbs

Question 5

Center of Mass:
xbar = ?
ybar = ?

Answer 5

xbar = My/M
ybar = Mx/M

Question 6

(Remember it is ybar)
(Remember delta is part of density)
Moment of Plate about x-axis:
Mx = ?

Answer 6

Mx = integral (ybar)dm
Mx =
Integral a to b
(delta/2)[f^2(x)-g^2(x)]dx

Question 7

(Remember it is xbar)
(Remember delta is part of density)
Moment of Plate about y-axis:
My = ?

Answer 7

My = integral (xbar)dm
My =
Integral a to b
( x)(delta)[f(x)-g(x)]dx

Question 8

Mass: M = ?

Answer 8

M = integral dm
Integral a to b
delta[f(x)-g(x)]dx

Question 9

Cosh(x) = ?

Answer 9

Cosh(x) =
(e^x + e^-x)/2

Question 10

Sinh(×) = ?

Answer 10

Sinh(x) =
(e^x - e^-x)/2

Question 11

Tanh(x) = ?

Answer 11

Tanh(x) = sinh(x)/cosh(x)

Question 12

Sech(x) = ?

Answer 12

Sech(x) = 1/cosh(x)

Question 13

Derivative of d/dx Sinh(u) = ?

Answer 13

Cosh(u) (du/dx)

Question 14

Derivative of d/dx cosh(u) = ?

Answer 14

sinh(u) du/dx

Question 15

Derivative of d/dx tanh(u) = ?

Answer 15

sech^2(u) du/dx

Question 16

Exp Change Formula = ?

Answer 16

y = y0e^(kt)

Question 17

Exp Change Shortcuts = ?

Answer 17

K = (ln(% change))/Time
Time = (ln(% change))/K
Halflife % change = 1/2
Double % change = 2

Question 18

Coth(x) = ?

Answer 18

Coth(x) = Cosh(x)/sinh(x)