Calculus 2 Unit 8 (Test 3)

• If f' is continuous on [a,b] , then the length of the curve y = f(x) is given by

  L = int from a to b sqrt ( 1 + (f'(x))^2 )dx

  or

  L = int from a to b sqrt ( 1 + (dy/dx)^2 ) dx
• Steps to find arc length

  1. Find the derivative of f(x) and square it
  2. Fill it into the arc length equation
  3. Do a u-sub of the inner part of the square root, integrate by parts, or any of the integration techniques
  4. Evaluate it at the bounds of the problem and solve
• Integration by parts formula

  int ( udv ) = uv - int (vdu )

  $\int_{ }^{ }udv\ =$$\ uv\ -\ \int_{ }^{ }vdu$
• Trigonometric Substitution, sqrt ( a^2 - x^2 )

  Side, sin(theta) = x/a -> x = a sin(theta)

  Use: 1 - sin^2(theta) = cos^2(theta)
• Trigonometric Substitution, sqrt (a^2 + x^2)

  Hypotenuse, tan(theta) = x/a -> x= a tan(theta)

  Use: 1 + tan^2(theta) = sec^2(theta)
• Trigonometric Substitution, sqrt (x^2 - a^2)

  Side, sec(theta) = x/a -> x= a sec(theta)

  Use: sec^2(theta) - 1 = tan^2(theta)
• sine = opp / hypo

  cosine = adj / hypo

  tangent = opp / adj
• sec = hypo / adj

  csc = hypo / opp

  cot = adj / opp
• If a curve has the equation x = g(y); c ≤ y ≤ d, & g'(y) is continuous, then:

  L = int from c to d sqrt ( 1 + (g'(y))^2 ) dy

  or

  L = int from c to d sqrt ( 1 + (dx/dy)^2 ) dy

  * usually used when the original f(x)'s derivative is not defined at the integral
• tan^2(x) + 1 = sec^2(x)
• sin^2(x) = 1 - cos^2(x)
• Area of a Surface of Revolution- About the x-axis

  $S\ =$$\ \int_{a}^{b}2\pi f\left(x\right)\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx$

  $S\ =$$\ \int_{c}^{d}2\pi y\sqrt{1+\left(\frac{dx}{dy}\right)^{2}}dy$
• Area of a Surface of Revolution- About the y-axis

  $S\ =$$\ \int_{a}^{b}2\pi x\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx$

  $S\ =$$\ \int_{c}^{d}2\pi g\left(y\right)\sqrt{1+\left(\frac{dx}{dy}\right)^{2}}dy$
• Steps to find the area of a surface of revolution

  1. Find the derivative of f(x) and square it
  2. Fill it into the arc length part of the equation and combine it with the 1 ( use LCD)
  3. Fill it into the formula, simplify, and evaluate the integral
• Rotate around the x-axis = use 'y'

  Rotate around the y-axis = use 'x'
• Arc Length Formula ( in terms of x)

  $A\ =$$\ \int_{a}^{b}\sqrt{1+\left(f'\left(x\right)\right)^{2\ }}dx$
• Arc Length Formula ( in terms of y)

  $A\ =$$\ \int_{c}^{d}\sqrt{1+\left(f'\left(y\right)\right)^{2\ }}dy$
• a^2 + 2ab + b^2 = (a + b)^2

  Factoring formula