# 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.

## Computing Arc Lengths

• 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.

## Understanding Surface Areas in Calculus

• 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.

## Calculating Surface Areas

• 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.

## Applications of Arc Lengths and Surface Areas

• 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.