Further Calculus: Arc lengths and surface areas

Understanding Arc Lengths in Calculus

The arc length is the distance along a curve between two points. In calculus, we use it to measure the length of a smooth, continuous curve defined by a function f(x) between two points a and b.
In mathematical terms, the length of the arc L between two points x=a and x=b on the curve y=f(x) is given by the integral of the square root of 1 plus the derivative of the function squared with respect to x, from a to b. This is represented as: L = ∫ from a to b sqrt[1 + (f’(x))^2] dx.
This should be read as “Integrate the square root of one plus the derivative of f(x) squared, with respect to x, from a to b”.
The derivation of this formula comes from the Pythagorean theorem, applied to infinitesimal segments of the curve.

To compute an arc length, you must first clearly define your curve function f(x), and decide on your interval [a, b].
Next, you must compute the derivative of the function f’(x).
Substitute f’(x) into the arc length formula and integrate over the interval from a to b.

For a solid of revolution formed by rotating a curve f(x), from x=a to x=b, around the x-axis, we define its surface area, A as the integral over the interval [a, b] of: 2π times the function f(x) times the square root of 1 plus the derivative of the function squared with respect to x.
The equation is: A = ∫ from a to b 2πf(x)sqrt[1 + (f’(x))^2] dx.
This formula is derived as the integral of the product of circumference 2πf(x) (as the function value f(x) serves as the radius when rotated around the x-axis) and infinitesimal arc length sqrt[1 + (f’(x))^2] dx.

To calculate surface area, you need to define your curve function f(x) and the interval of rotation [a, b].
Then, compute the derivative of the function f’(x), and substitute f’(x) and f(x) into the surface area formula.
Finally, take the integral over the interval from a to b.

The concepts of arc length and surface area have a broad range of applications, for example, in physics and engineering for determining paths of least distance (geodesics), measurements of natural phenomena, or in the design and manufacture of curved shapes.
These applications underline the importance of understanding how to compute arc lengths and surface areas exactly.