- 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 :
- For :
- For :
- 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 formula when curve is better expressed as a function of
- Use formula when curve is better expressed as a function of
Let Calculate the arc length of the graph of over the interval Round the answer to three decimal places.
Let Calculate the arc length of the graph of over the interval 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 -axis:
- For rotation around -axis:
- For rotation around -axis:
- Frustum of a cone:
- Used to approximate small sections of the surface
- Lateral surface area: , 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 -axis formula when curve is better expressed as
- Use -axis formula when curve is better expressed as
Let over the interval Find the surface area of the surface generated by revolving the graph of around the Round the answer to three decimal places.
Let over the interval Find the surface area of the surface generated by revolving the graph of around the