FP2 2.2 - Arc length and surface area

Add small sloping lengths, rather than horizontal widths

Can a sphere have exactly the same area as the curved cylinder fitted tightly around it? Its curved profile and the cylinder's straight profile have different lengths, but each produces area by sweeping circles of changing radius. We will derive the two areas and explain why they agree.

Join nearby curve points by a short straight segment. Pythagoras gives $\Delta s\approx\sqrt{(\Delta x)^2+(\Delta y)^2}$. In the limit, for a curve $y=f(x)$,

when integrating with increasing $x$. The slope measures how much longer the sloping piece is than its horizontal projection.

For a piecewise smooth curve, add the lengths of its pieces. Choose an increasing interval of the integration variable so that a geometric length is positive.

When $x$ is given as a function of $y$, interchange the coordinate roles and integrate $\sqrt{1+(\mathrm{d}x/\mathrm{d}y)^2}$ with respect to increasing $y$. This follows from the same distance calculation; it is not printed as a separate formula.

Keep the curve and lower endpoint fixed; change only the upper endpoint to $\ln3$.

These particular curve lengths are derived values, not entries in the formula booklet.

Extend the same curve again.

Use both coordinate derivatives for a parametric curve

For $x=x(t),\ y=y(t)$, the small horizontal and vertical changes are $x'(t)\,\mathrm{d}t$ and $y'(t)\,\mathrm{d}t$. Pythagoras combines them into a speed. A negative coordinate derivative does not make that speed negative.

Consider $x=2(t-\sin t),\ y=2(1-\cos t)$, $0\le t\le2\pi$, one arch of a cycloid. Its derivatives both vanish at each end, but its speed is continuous and its length is finite. The diagram identifies the half and whole traces.

[DIAGRAM: asset_name: Half and whole cycloid arches; asset_slug: 9fm0-fp293-cycloid-arc; caption: Parameter bounds select a half arch or a full arch without eliminating the parameter; description: An equal-scale Cartesian plot shows x equals two times the quantity t minus sin t, and y equals two times the quantity one minus cos t, for t from zero to two pi. The first half, zero to pi, is blue and the second half, pi to two pi, is sage. Endpoints (zero, zero), (two pi, four) and (four pi, zero) are labelled in dark ink with their parameter values. The legend is outside the curve. A note gives half-arch length eight and full-arch length sixteen.]

The speed simplifies because $(1-\cos t)^2+\sin^2t=2-2\cos t=4\sin^2(t/2)$. Keep its absolute value until the parameter interval fixes the sign.

Keep the equations and lower bound fixed; change only the upper parameter bound to $2\pi$.

The cycloid lengths are derived, not printed. If a parametrisation travels over the same curve portion repeatedly, use only the trace requested; the speed integral counts each traversal.

Use a shorter first portion of the same arch.

Include both radial change and angular travel

For a polar curve, increasing the angle moves the point even if its radius stays constant. Differentiate $x=r\cos\theta,\ y=r\sin\theta$; adding the squared derivatives cancels the cross-terms and gives $(r')^2+r^2$.

For $r=2(1+\cos\theta)$, the upper half is traced once by $0\le\theta\le\pi$, ending at the pole. Its radius is nonnegative. The pole endpoint causes no difficulty for length; check the speed rather than dividing by $r$.

[DIAGRAM: asset_name: Upper-half and shorter cardioid traces; asset_slug: 9fm0-fp293-cardioid-traces; caption: The angle interval decides how much of the nonnegative-radius curve is measured; description: An equal-scale Cartesian cardioid r equals two times the quantity one plus cos theta is shown. The first segment, theta from zero to pi over two, is sage; the remainder of the upper half to theta pi is blue; the lower half is a thin grey dashed reference curve. Points (four, zero), (zero, two) and the pole are labelled with angles zero, pi over two and pi. A vertical dashed ray marks theta pi over two. Legends are outside the plotting area. Notes give upper-half length eight and first-segment length four root two.]

The square root will contain $\cos^2(\theta/2)$. Its sign is known on the selected upper-half interval, but would change during a full angular turn.

Keep the radius function and starting angle fixed; change only the upper angle to $\pi/2$.

These particular lengths are not in the formula booklet. A full cardioid would require $|\cos(\theta/2)|$, split where its sign changes, rather than the upper-half simplification over the whole turn.

Stop at a different admissible ray.

Multiply each sloping length by the circumference it sweeps

A short curve segment at distance $\rho$ from an axis sweeps a narrow frustum band. Its area approaches circumference times slant length, $2\pi\rho\,\mathrm{d}s$, for a full turn. Thus curved surface area is accumulated from $2\pi\rho\,\mathrm{d}s$, rather than from the disc area $\pi\rho^2$ used for volume. This unifying band rule is derived here, not printed separately in that form.

The radius is a distance from the stated axis. About the $x$-axis it is $|y|$; about the $y$-axis it is $|x|$. A curve may sweep an open-ended surface: end discs are not included unless the question asks for them.

The printed form uses the positive radius $y$. If the curve is below the axis, use $|y|$. About the $y$-axis, replace the radius factor by $|x|$, keeping the same length element. These distance/axis extensions follow from the band reasoning; they are not printed as separate formulas.