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Parametric Integration

Parametric Integration

Many curves we integrate come in the form y = f (x). For most curves, this is fine, but it is not always possible or convenient to write it like this. It is in this scenario where parametric coordinates are useful.

Recap of parametric coordinates

In this scenario, let us introduce a 'dummy' variable, usually denoted as t. We call this a dummy variable as it is an abstract concept that assigns a value to either an x or y coordinate, and isn't plotted.

This means that instead of having a function of the form y = f (x), we represent a curve by y (t) = g (t), x (t) = h (t), where h and t are functions that describe the change of the x and y coordinates respectively.

A curve is described by y (t) = 2 (t), x (t) = 2 (t), 0 <t <2π.

Expressing the parametric curve as (x (t)) ² + (y (t)) ² = (2cos (t)) ² + (2sin (t)) 2 = 4 (cos² (t) + sin² (t)) = 4, we see that it actually describes a circle of radius 4, or x² + y² = 4.

Why does parametric integration work?

Normally, we expect to evaluate an integral of the form ∫y (x) dx; however, we need to change this because our curve is not in the form y (x). We use a modified version of the Chain Rule. We can replace dx with (you can think of this as the dt's cancelling. While this is not technically how they work, as dx / dt is not strictly a fraction, we can treat it as one for operational purposes). This gives an integral of the form.

We must also remember to do with parametric integrals is switch limits. Suppose we have an integral of the form. We must also switch the limits, which results in the integral being given as, where

.

Examples of parametric integration

At first glance, this can be a tricky topic to get your head around, so let's walk through a couple of examples to try and consolidate what we have said so far.

Parametric Integration - Key takeaways

  • The formula for parametric integration is given as

  • We must remember to switch limits when changing from x coordinates to t coordinates.

Frequently Asked Questions about Parametric Integration

Integrate parametrically using the formula  ∫y(t).dx(t/)dt.dt

Use the inverse formula for x(t) to find the t limits from the x limits.

Parametric integration works as we introduce a dummy variable which essentially shifts our curve to a different plane where it is easier to integrate.

Final Parametric Integration Quiz

Question

Why are some curves defined parametrically?

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Answer

Parametric equations are used as sometimes we cannot describe a curve normally, or it’s easier to write them parametrically.

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Question

Explain the term ‘dummy variable’

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Answer

Dummy variable refers to a variable that is introduced that helps describe the various x and t coordinates.

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Question

How do we change from t coordinates in the limit to x coordinates?

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Answer

Fill in the values of t into the function (x(t))

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Question

A curve is defined as x(t)=t²-4, y(t)=4t. Find the area under the curve which is enclosed by the line x=2, x=8 and the x axis.

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Answer

1344

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Question

A curve is defined as x(t)=t²+3, y(t)=1/t, t>0. Find the area under the curve which is enclosed by the line x=4, x=7 and the x axis.

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Answer

6

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Question

A curve is defined as x(t)=t³, y(t)=4t. Find the area under the curve which is enclosed by the line x=1, x=9 and the x axis. 


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Answer

19680

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Question

A curve is defined as x(t)=t³-2, y(t)=6t. Find the area under the curve which is enclosed by the line x=4, x=11 and the x axis.

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Answer

129465/2

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Question

A curve is defined as x(t)=3-t, y(t)=4t². Find the area under the curve which is enclosed by the line x=-4, x=9 and the x axis.

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Answer

-260/3

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Question

An ellipse is defined parametrically as x(t)=3sin(t), y(t)=4cos(t), 0<t<2π. Find the area of the ellipse integrating parametrically.

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Answer

12π

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Question

An ellipse is defined parametrically as x(t)=6sin(t), y(t)=2cos(t), 0<t<2π. Find the area of the ellipse integrating parametrically.

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Answer

12π

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Question

An ellipse is defined parametrically as x(t)=4sin(2t), y(t)=9cos(2t), 0<t<π. Find the area of the ellipse integrating parametrically.

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Answer

36π

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Question

A circle is defined parametrically as x(t)=7sin(4t), y(t)=7cos(4t), 0<t<π/2. Find the area of the circle integrating parametrically.

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Answer

49π

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