where r, θ are polar coordinates, and p is the perpendicular drawn from a fixed point to the tangent. The integrals are to be understood as line integrals taken along the curve. When the same integrals are taken between limits which correspond to two points of the curve, in the sense of line integrals along the arc between the points, they represent the area bounded by the arc and the terminal radii vectores.
(iv.) The volume enclosed by a surface which is generated by the revolution of a curve about the axis of x is expressed by the formula
π ∫ y2 dx,
and the area of the surface is expressed by the formula
2π ∫ y ds,
where ds is the differential element of arc of the curve. When the former integral is taken between assigned limits it represents the volume contained between the surface and two planes which cut the axis of x at right angles. The latter integral is to be understood as a line integral taken along the curve, and it represents the area of the portion of the curved surface which is contained between two planes at right angles to the axis of x.
(v.) When we use curvilinear coordinates ξ, η which are conjugate functions of x, y, that is to say are such that
∂ξ/∂x = ∂η/∂y and ∂ξ/∂y = −∂η/∂x,
the Jacobian ∂(ξ, η)/∂(x, v) can be expressed in the form
| ( | ∂ξ | )2 + ( | ∂η | )2, |
| ∂x | ∂x |