The Main Idea 

• Arc length represents the distance along a curve
• Requires smooth functions (continuous derivatives)
• Approximated using line segments, then taking the limit
• Two main formulas:
  • For $y = f(x)$:
    • $\text{Arc Length} = \int_a^b \sqrt{1 + [f'(x)]^2} dx$
  • For $x = g(y)$:
    • $\text{Arc Length} = \int_c^d \sqrt{1 + [g'(y)]^2} dy$
• Smoothness requirement:
  • Function must be differentiable with a continuous derivative
• Derivation approach:
  • Partition the interval
  • Approximate curve with line segments
  • Use Pythagorean theorem for segment length
  • Sum segment lengths and take the limit
• Choice of formula:
  • Use $y = f(x)$ formula when curve is better expressed as a function of $x$
  • Use $x = g(y)$formula when curve is better expressed as a function of $y$

The Main Idea 

• Surface area is the total area of the outer layer of an object
• For surfaces of revolution, we use calculus to find the area
• The method extends concepts from arc length calculations
• Two main formulas:
  • For rotation around $x$-axis:
    • $\text{Surface Area} = \int_a^b 2\pi f(x)\sqrt{1 + [f'(x)]^2} dx$
  • For rotation around $y$-axis:
    • $\text{Surface Area} = \int_c^d 2\pi g(y)\sqrt{1 + [g'(y)]^2} dy$
• Frustum of a cone:
  • Used to approximate small sections of the surface
  • Lateral surface area: $S = \pi(r_1 + r_2)l$, where l is slant height
• Derivation approach:
  • Partition the interval
  • Approximate surface with frustums
  • Sum frustum areas and take the limit
• Smooth function requirement:
  • Function must be differentiable with a continuous derivative
• Choice of formula:
  • Use $x$-axis formula when curve is better expressed as $y = f(x)$
  • Use $y$-axis formula when curve is better expressed as $x = g(y)$