Arc Length of Curves

General Formula for Arc Length

The arc length of a curve can be calculated using integral formulas, depending on whether the curve is given as y = f(x) or x = g(y). The arc length, L, between two points is:

• For y = f(x):

• For x = g(y):

These formulas are derived from the Pythagorean theorem applied to infinitesimal segments of the curve.

Example: Arc Length for x = cos(y)

Given the curve x = \cos(y) for 0 \leq x \leq \frac{1}{2}, the arc length can be set up using both dx and dy:

• Using dx:

• Using dy:

To use these formulas, derivatives must be computed:

Substituting these derivatives gives:

The limits for y are found by solving x = \cos(y) for y:

• When x = 0, y = \arccos(0) = \frac{\pi}{2}

• When x = \frac{1}{2}, y = \arccos(\frac{1}{2}) = \frac{\pi}{3}

Final integral expressions:

Surface Area of Solids of Revolution

General Formula for Surface Area

The surface area of a solid formed by revolving a curve about an axis is given by:

• About the y-axis (using dx):

• About the y-axis (using dy):

These formulas account for the circumference of each infinitesimal ring generated by the revolution.

Example: Surface Area for x = \sqrt{y+5}

Given x = \sqrt{y+5} for \sqrt{5} \leq x \leq 3, the surface area can be set up using both dx and dy:

• Rewrite as y = x^2 - 5 and compute

• Substitute into the formula:

• Surface area integral:

Alternatively, using dy:

• Surface area integral:

• Simplified:

Example: Surface Area for y = 3x^2

Given y = 3x^2, compute and substitute:

• Surface area integral:

Summary Table: Arc Length and Surface Area Formulas

Key Points and Applications

• Arc length integrals are used to find the length of curves, important in geometry and physics.

• Surface area integrals are used to find the area of solids of revolution, relevant in engineering and design.

• Choosing between dx and dy depends on the form of the function and the given limits.

• Always compute the correct derivative and adjust the limits if changing variables.