Summary of Arc Length of a Curve and Surface Area

The arc length of a curve can be calculated using a definite integral.
The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of $y.$
The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.
The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals.

Key Equations

Arc Length of a Function of $x$
$\text{Arc Length}={\displaystyle\int }_{a}^{b}\sqrt{1+{\left[{f}^{\prime }(x)\right]}^{2}}dx$
Arc Length of a Function of $y$
$\text{Arc Length}={\displaystyle\int }_{c}^{d}\sqrt{1+{\left[{g}^{\prime }(y)\right]}^{2}}dy$
Surface Area of a Function of $x$
$\text{Surface Area}={\displaystyle\int }_{a}^{b}(2\pi f(x)\sqrt{1+{({f}^{\prime }(x))}^{2}})dx$

arc length: the arc length of a curve can be thought of as the distance a person would travel along the path of the curve

frustum: a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base

surface area: the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces