For example, when we have to integrate a function ƒ(x, y) over the area within a circle given by x2 + y2 = a2, and we introduce polar coordinates so that x = r cos θ, y = r sin θ, we find that r is the value of the Jacobian, and that all points within or on the circle are given by a ≥ r ≥ 0, 2π ≥ θ ≥ 0, and we have
| ∫a−a dx ∫ | √(a2−x2) | ƒ(x, y) dy = ∫a0 dr ∫2π0 ƒ(r cos θ, r sin θ) r dθ. |
| −√(a2−x2) |
If we have to integrate over the area of a rectangle a ≥ x ≥ 0, b ≥ y ≥ 0, and we transform to polar coordinates, the integral becomes the sum of two integrals, as follows:—
∫a0 dx ∫b0 ƒ(x, y) dy = ∫0tan−1b/a dθ ∫0a sec θ ƒ(r cos θ, r sin θ) r dr
+ ∫1/2πtan−1b/a dθ ∫0b cosec θ ƒ(r cos θ, r sin θ) r dr.
55. A few additional results in relation to line integrals and multiple integrals are set down here.
(i.) Any simple integral can be regarded as a line-integral taken along a portion of the axis of x. When a change of Line Integrals and Multiple Integrals. variables is made, the limits of integration with respect to the new variable must be such that the domain of integration is the same as before. This condition may require the replacing of the original integral by the sum of two or more simple integrals.
(ii.) The line integral of a perfect differential of a one-valued function, taken along any closed curve, is zero.
(iii.) The area within any plane closed curve can be expressed by either of the formulae
∫ ½ r2 dθ or ∫ ½ p ds,