calc chapter 8

polar length equation

integral from a to b of the square root of r squared plus d r / d theta squared d theta

polar area equation

the integral from a to b of ½ r squared d theta (NO SQUARE ROOT)

parametric length equation

the integral from a to b of the square root of dx/dt squared plus dy/dt squared dt

regular Cartesian length equation

the integral from a to b of the square root of 1 plus f prime squared dx

surface area equation

the integral from a to b of 2 pi r times the square root of 1 plus f prime squared dx

cylindrical shells equation

the integral from a to b of 2 pi r h dx (THIS IS THE ONLY NEW ONE THAT CAN HAVE DY)

discs and annuli equation

the integral from a to b of pi R squared minus r squared dx (for discs just do pi r squared) (REMEMBER NO 2 PI JUST PI) (THIS CAN HAVE DY)

cardoid equation

a+a sin theta

limacon equation

a+b sin theta and you can figure this out by plugging in numbers but when absolute value of a is greater than that of b, there’s no loop the only thing I can think of to memorize is “WITH loop, the less than symbol points WEST”

rose equation

a sin b theta, if the absolute value of b is odd, that’s the number of leaves, if its even, then thats half the number of leaves. abs a is length of leaf

circle equation

r = a, r = a sin theta , if a is greater than 0, the circle is in the quadrants where sin/cos is positive depending on which Is used, also symmetric about y axis for sin and x axis for cos.

spiral equation

r=a theta

remember to state domain in optimization problem

sure

when determining global max/min, remember you have to include endpoints if the interval is closed

and for open, use lonely critical points approach