Arc Length of a Curve and Surface Area: Fresh Take

  • Calculate the length of a curve described by y=f(x) from one point to another
  • Find the length of a curve defined by x=g(y) from one point to another
  • Calculate the total surface area of a solid formed by rotating a curve around an axis

Arc Lengths of Curves

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):
      • Arc Length=ab1+[f(x)]2dx
    • For x=g(y):
      • Arc Length=cd1+[g(y)]2dy
  • 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

Let f(x)=(43)x3/2. Calculate the arc length of the graph of f(x) over the interval [0,1]. Round the answer to three decimal places.

Let f(x)=sinx. Calculate the arc length of the graph of f(x) over the interval [0,π]. Use a computer or calculator to approximate the value of the integral.

Area of a Surface of Revolution

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:
      • Surface Area=ab2πf(x)1+[f(x)]2dx
    • For rotation around y-axis:
      • Surface Area=cd2πg(y)1+[g(y)]2dy
  • Frustum of a cone:
    • Used to approximate small sections of the surface
    • Lateral surface area: S=π(r1+r2)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)

Let f(x)=1x over the interval [0,12]. Find the surface area of the surface generated by revolving the graph of f(x) around the x-axis. Round the answer to three decimal places.

Let g(y)=9y2 over the interval y[0,2]. Find the surface area of the surface generated by revolving the graph of g(y) around the y-axis.