Calculus with Parametric Curves: Learn It 3

Integrals Involving Parametric Equations

Now that we can find derivatives of parametric curves, let’s tackle finding the area under these curves. Consider the cycloid defined by [latex]x(t) = t - \sin t[/latex] and [latex]y(t) = 1 - \cos t[/latex]. How do we find the area of the shaded region below?

A series of half circles drawn above the x-axis with x intercepts being multiples of 2π. The half circle between 0 and 2π is highlighted. On the graph there are also written two equations: x(t) = t – sin(t) and y(t) = 1 – cos(t).
Figure 7. Graph of a cycloid with the arch over [latex]\left[0,2\pi \right][/latex] highlighted.

To find the area under a parametric curve defined by [latex]x = x(t)[/latex] and [latex]y = y(t)[/latex] for [latex]a \le t \le b[/latex], we’ll use the familiar rectangle approximation method.

Start by partitioning the interval [latex][a, b][/latex] into [latex]n[/latex] subintervals: [latex]t_0 = a < t_1 < t_2 < \cdots < t_n = b[/latex].

A curved line is drawn in the first quadrant. Below it are a series of rectangles marked that begin at the x-axis and reach up to the curved line; the rectangle’s height is determined by the location of the curved line at the leftmost point of the rectangle. These lines are noted as x(t0), x(t1), …, x(tn).
Figure 8. Approximating the area under a parametrically defined curve.

For each rectangle:

  • Height: [latex]y(\overline{t}_i)[/latex] for some [latex]\overline{t}_i[/latex] in the [latex]i[/latex]th subinterval
  • Width: [latex]x(t_i) - x(t_{i-1})[/latex]

The area of the [latex]i[/latex]th rectangle becomes:

[latex]{A}_{i}=y\left(x\left({\overline{t}}_{i}\right)\right)\left(x\left({t}_{i}\right)-x\left({t}_{i - 1}\right)\right)[/latex].

The total approximate area is:

[latex]{A}_{n}=\displaystyle\sum _{i=1}^{n}y\left(x\left({\overline{t}}_{i}\right)\right)\left(x\left({t}_{i}\right)-x\left({t}_{i - 1}\right)\right)[/latex].

To convert this into a form we can integrate, multiply and divide by [latex]\Delta t = t_i - t_{i-1}[/latex]:

[latex]{A}_{n}=\displaystyle\sum _{i=1}^{n}y\left(x\left({\overline{t}}_{i}\right)\right)\left(\frac{x\left({t}_{i}\right)-x\left({t}_{i - 1}\right)}{{t}_{i}-{t}_{i - 1}}\right)\left({t}_{i}-{t}_{i - 1}\right)=\displaystyle\sum _{i=1}^{n}y\left(x\left({\overline{t}}_{i}\right)\right)\left(\frac{x\left({t}_{i}\right)-x\left({t}_{i - 1}\right)}{\Delta t}\right)\Delta t[/latex].

Taking the limit as [latex]n[/latex] approaches infinity gives:

[latex]A=\underset{n\to \infty }{\text{lim}}{A}_{n}={\displaystyle\int }_{a}^{b}y\left(t\right){x}^{\prime }\left(t\right)dt[/latex].

As [latex]n[/latex] approaches infinity, the change in [latex]x[/latex] over smaller and smaller time intervals becomes the instantaneous rate of change [latex]x'(t)[/latex]. This transformation relies on the Mean Value Theorem.

Mean Value Theorem
[latex]\\[/latex]
If [latex]f[/latex] is continuous on [latex][a,b][/latex] and differentiable on [latex](a,b)[/latex], then there exists at least one point [latex]c \in (a,b)[/latex] such that:

[latex]f'(c) = \frac{f(b) - f(a)}{b - a}[/latex]

This theorem guarantees that the average rate of change equals the instantaneous rate of change at some point in the interval.

The preceding result leads to the following theorem.

theorem: area under a parametric curve

For a non-self-intersecting plane curve defined by:

  • [latex]x = x(t)[/latex], [latex]y = y(t)[/latex] where [latex]a \le t \le b[/latex]
  • [latex]x(t)[/latex] is differentiable

The area under the curve is:

[latex]A = \int_a^b y(t) \cdot x'(t) dt[/latex]

Important: This formula assumes the curve doesn’t cross itself and that [latex]x(t)[/latex] increases as [latex]t[/latex] increases from [latex]a[/latex] to [latex]b[/latex].

Find the area under the curve of the cycloid defined by the equations

[latex]x\left(t\right)=t-\sin{t},y\left(t\right)=1-\cos{t},0\le t\le 2\pi[/latex].

 

Watch the following video to see the worked solution to the example above.

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You can view the transcript for this segmented clip of “7.2 Calculus of Parametric Curves” here (opens in new window).