Unit 8

Question 1

Disk method

Answer 1

π ∫ (r(x))² dx

• only when:
  about x-axis in terms of x
  about y-axis in terms of y
• no hole/gap

Question 2

Washer method

Answer 2

π ∫ (R² - r²) dx

• only when:
  about x-axis in terms of x (top - bottom)
  about y-axis in terms of y (right - left)
• hole/gap

Question 3

Shell method

Answer 3

2π ∫ r(x)h(x) dx

• only when:
  about x-axis in terms of y
  about y-axis in terms of x
  (mismatch)
• cylinders
• h(x) is what forms outside of 3D shape, nothing inner

Question 4

Volume by known cross sections

Answer 4

∫ (area of shape) dx

• cross sections perpendicular to:
  x-axis: in terms of x (top - bottom)
  y-axis: in terms of y (right - left)

Question 5

Arc Length

Answer 5

S = ∫ (1 + (dy/dx)²)^1/2

Question 6

Distance (ds) between two points on a graph

Answer 6

ds = [(dx)² + (dy)²]^1/2

(dy)² + (dx)² = (ds)²