# Parametric Area

Suppose a parametric curve in \(\mathbb{R}^2\) is given by \(x(t)\), \(y(t)\), where the parameter \(t\) ranges over some given time interval. This curve may bound a region of the Cartesian plane (in particular, when the curve is closed and has no self-intersections), and one may wish to compute the **area** of this region. This can be done using the formula below:

If \(x(t)\), \(y(t)\), \(t\in [a,b]\) describes a parametric curve bounding a region of the Cartesian plane, the area of this region is \[\int y \, dx = \int_{a}^{b} y(t) x'(t) \, dt.\]

## Derivation Of Formula

Suppose \(x(t)\), \(y(t)\), \(t\in [0,1]\) describes a parametric curve that bounds a region \(R \subset \mathbb{R}^2\). Intuitively, one can approximate the area of \(R\) using rectangles; for an infinitesimal change in \(x\), from \(x\) to \(x+dx\), the portion of \(R\) contained between the "fenceposts" \(x\) and \(x+dx\) is approximately a rectangle, with height \(y\) and width \(dx\).

Since the area of this rectangle is \(y \, dx\), integrating this element over all possible \(x\)-values should give the area of \(R\). Thus, the area is precisely \[\int y \, dx = \int_{0}^{1} y(t) x'(t) \, dt.\] This proof can be made rigorous using Green's theorem.

If the argument above had been carried out swapping the roles of \(x\) and \(y\), one would obtain the formula \[\int x \, dy = \int_{0}^{1} x(t) y'(t) \, dt.\] This makes sense because the curve must be *closed* in order for it to bound a region, i.e. one must have \((x(0), y(0)) = (x(1), y(1))\). Then, one may compute \[\int x \, dy + \int y \, dx = \int_{0}^{1} [x(t) y'(t) + y(t) x'(t)] \, dt = \int_{0}^{1} (x(t) y(t))' \, dt = x(1) y(1) - x(0) y(0) = 0.\] Thus, the two formulas for area are equal *up to a sign*.

This illustrates the idea that this formula actually gives *signed area*, meaning that the value of the integral \[\int y\, dx\] is positive or negative depending on whether the bounded region lies to the left or right of the curve. For example, note that the regions bounded by closed curves \((x(t), y(t))\) and \((y(t), x(t))\) have the same unsigned area (which equals absolute value of the signed area), since one curve is just the reflection of the other over the line \(y = x\); but their signed areas sum to zero.

## Examples And Problems

What is the area of a circle with radius \(r\)?

Parametrize the circle as \((x(t), y(t)) = (r\cos(t), r\sin(t))\), where \(t\in [0,2\pi]\). From the formula for area bounded by a parametric curve, we have \[\text{Area}(\text{circle of radius } r) = \int_{0}^{2\pi} r \cos(t) \cdot (r\sin(t))' \, dt = \int_{0}^{2\pi} r^2 \cos^2 (t) \, dt = \pi r^2. \]

A

cycloidis the parametric curve given by equations \[x(t) = t-\sin(t),\] \[y(t) = 1-\cos(t).\]From the image, one notes that the cycloid consists of many congruent arches, traced as \(t\) ranges over the real numbers. What is the area bounded by one arch of the cycloid?

One arch is traced out in its entirety as \(t\) ranges over \([0,2\pi]\). We compute \[\int y \, dx = \int_{0}^{2\pi} (1-\cos(t))^2 \, dt = \left[ \frac{3}{2} x - 2\sin(x) + \frac{1}{4} \sin(2x) \right] \Big\vert_{0}^{2\pi} = 3\pi.\]

**Cite as:**Parametric Area.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/parametric-equations-area-basic/