Arc Length and the Area of a Surface of Revolution
Set up the integrals that measure the length of a plane curve and the area swept when it is spun about an axis.
What a learner can do afterwards
- Derive the arc length integrand from a limit of straight-line approximations
- Compute the arc length of a curve given in Cartesian and in parametric form
- Set up and evaluate a surface-of-revolution area integral about each axis
1 · Read
Arc length starts from Pythagoras. A tiny piece of curve is nearly straight, with length from dx and dy. Factoring out dx gives the famous integrand: the square root of (1 plus (dy/dx) squared). Integrating adds the pieces exactly. So the formula comes from adding short straight segments and passing to a limit.
Straight lines keep the integral friendly. For y is (3/4)x, the slope is 3/4, so the integrand is the constant 5/4. From 0 to 4 that gives 5, and from 0 to 8 it gives 10. A horizontal line has zero slope, so y is 5 from 2 to 9 has length 9 minus 2, which is 7. Constant integrand means length is just rate times width.
Revolving a curve sweeps out bands, and each band unwraps to nearly a rectangle: circumference times width. The circumference is 2 pi times the radius, and the width is the arc length element. About the x-axis the radius is the function value f(x), and about the y-axis it is the distance to that axis, g(y). Only the integral with both the 2 pi f(x) factor and the square root is the surface area.
Parametrise curves you cannot write as y is f(x), then integrate speed: the square root of ((dx/dt) squared plus (dy/dt) squared). Sanity check answers by revolving a straight segment, which gives a cone or cylinder with geometry you already know. The same idea covers wider ground with no new machinery.
Build length from tiny straight pieces, spin bands for area, and parametrise when y will not cooperate.
More to read
6.4 Arc Length of a Curve and Surface Area
Arc length starts from Pythagoras: a tiny piece of curve is nearly straight, with length sqrt(dx² + dy²). Factoring out dx gives the famous integrand sqrt(1 + (dy/dx)²). Integrating adds the pieces exactly. The same idea in parametric form integrates speed instead.
2.4 Arc Length of a Curve and Surface Area
Revolving a curve sweeps out bands, and each band unwraps to nearly a rectangle: circumference times width. The circumference is 2π times the radius, and the width is the arc length element. That product, integrated, is the whole surface area formula. About the x-axis the radius is y; about the y-axis it is x.
7.2 Calculus of Parametric Curves
Parametric curves describe motion directly: x(t) and y(t) tell you where you are at time t. Their speed sqrt((dx/dt)² + (dy/dt)²) integrates to arc length with no need to solve for y as a function of x. Surfaces from parametric curves work the same way, with radius read off the coordinate that measures distance to the axis.
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