Calc 2 Useful Formulas

Area of a square

A = s²

Area of semicircle

A = (pi/8)d²

Area of an equilateral triangle

A = (sqrt of 3)/4 x s²

Disk Formula

V = (pi) integral of r(x)²

Washer method

V = (pi) integral of R² - r²

Shell method

V = (2pi) integral of r(x) times h(x)

Decision rule

Perpendicular → disk/washer
Parallel → shells

Arc Length

L = integral of sqrt of 1 + y’²

Surface Area

S = (2pi) integral of y multiplied by the sqrt of 1 + y’²

Surface area tip

(Arc length)(radius) = surface area

General work

W = integral of F(x) dx

Hooke’s Law

F = kx

W for springs

W = (1/2) kx² (From 0 to x)

Lifting rope/cable

Weight of small piece = (density)(g)(length)
Distance lifted = (top - y)

Pumping fluids

W = integral of (density x g) (h-y) (A(y)) dy

Work tip

Work = weight x distance

sinh(x)

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

cosh (x)

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

(sinhx)’

cosh (x)

(cosh (x))’

sinh (x)

cosh²x - sinh²x

cosh²x -sinh²x = 1

u-sub

Pick u = inner function
Compute du
Replace everything
Change bounds if definite
Integrate
Substitute back

Common forms of u-substitution

u = (ax + b)
u = something inside a root
u = exponent or trig argument

Radius

The distance to axis of rotation

Height

the difference between the two curves