Suggested languages for you:
|
|

## All-in-one learning app

• Flashcards
• NotesNotes
• ExplanationsExplanations
• Study Planner
• Textbook solutions

# 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.

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?

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

Show question

Question

Explain the term ‘dummy variable’

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

Show question

Question

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

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

Show question

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.

1344

Show question

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.

6

Show question

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.

19680

Show question

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.

129465/2

Show question

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.

-260/3

Show question

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.

12π

Show question

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.

12π

Show question

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.

36π

Show question

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.

49π

Show question

60%

of the users don't pass the Parametric Integration quiz! Will you pass the quiz?

Start Quiz

## Study Plan

Be perfectly prepared on time with an individual plan.

## Quizzes

Test your knowledge with gamified quizzes.

## Flashcards

Create and find flashcards in record time.

## Notes

Create beautiful notes faster than ever before.

## Study Sets

Have all your study materials in one place.

## Documents

Upload unlimited documents and save them online.

## Study Analytics

Identify your study strength and weaknesses.

## Weekly Goals

Set individual study goals and earn points reaching them.

## Smart Reminders

Stop procrastinating with our study reminders.

## Rewards

Earn points, unlock badges and level up while studying.

## Magic Marker

Create flashcards in notes completely automatically.

## Smart Formatting

Create the most beautiful study materials using our templates.