Math 1B Mini Exam & Midterm I

disk volume formula

Volume = π r² Δx

washer volume formula

Volume = (π R² - π r²) Δx

cylindrical shell volume formula

2π r h Δx

disk method slices

vertical

washer method slices

vertical

cylindrical shell method slices

horizontal

disk method traits

• solid (not hollow)

• dx for HORIZONTAL axis of rotation

• dy for VERTICAL axis of rotation

washer method traits

• hollow

• dx for HORIZONTAL axis of rotation

• dy for VERTICAL axis of rotation

cylindrical shell method traits

• either solid or hollow

• dy for VERTICAL axis of rotation

• dx for HORIZONTAL axis of rotation

probability definition

probability density function

probability that x lies in an interval [a, b]

median value of x

number M such that

mean value of x or the expected value of x

Comparison Theorem

0 ≤ f (x) ≤ g(x)

Comparison Theorem warnings

1/x converge or diverve?

diverge

1/x² converge or diverge?

converge

1/x³ converge or diverge?

converge (check)

trapezoidal sum formula

left- and right-hand sum formulas

Error in Numerical Approximations (trapezoid & midpoint)

mid point sum formula

Error Bounds in Numerical Approximations (left and right hand sums)

work formula

work = mgh

limit definition of an integral

∆x

x_k

p-integral rule

if p = 1

function diverges

product rule

triangle inequality

circle formula

add later: common derivatives, similar triangles, improper integrals, & steps for common problems