Arc Length of a Curve and Surface Area

The arc length formula

Symmetry and special cases

Surface area of revolution

Work and the physical template

For a curve y = f(x) on [a, b], which integrand gives the arc length?

The arc length formula applied to the line y = mx on [0, a] gives what length?

A piece of curve of length ds at distance r from the axis of rotation sweeps what band area?

Rotating the semicircle y = sqrt(r^2 - x^2) about the x axis, what surface area integral gives the sphere?

Rotating y = 1/x on [1, infinity) about the x axis gives a surface with infinite area but finite volume. What does this demonstrate?

For which curve does the arc length integral on a finite interval evaluate in elementary closed form?

For y = x on [0, 1], the arc length integral is the integral of sqrt(2) dx. What length results? Answer like sqrt(2).

The cone's surface area from rotation is pi r times the slant height, a single letter usually used for it. What letter? Answer with one lowercase letter.

The sphere's surface area integral 2 pi integral of r dx from -r to r evaluates to what multiple of pi r^2? Answer with a number.

The arc length element ds equals sqrt of 1 plus a squared derivative. As two letters followed by prime notation, what is that derivative? Answer like f'.