Study materials›Calculus II›Parametric Equations and Polar Coordinates
🌀CALCULUS II STUDY GUIDE

Parametric Equations and Polar Coordinates

Trace curves, calculate slopes, areas, and arc lengths in parametric and polar form.

8 key ideas8 flashcards5 practice questions
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EXAM SCOPEOpenStax Calculus Volume 2, parametric equations and polar coordinates

This review follows the current course framework.

Checked against OpenStax Calculus Volume 2 ↗
TWO CURVE LANGUAGES

Track direction and repeated points before calculating.

x(t), y(t)

parameter sets position and direction

r(θ)

radius and angle name a point

trace interval

count each arc once

SMART NOTES

Key ideas to know

Start with the relationships between ideas. Then close the notes and explain each one from memory.

  1. 01

    Parametric equations give x and y as functions of a third variable.

  2. 02

    A curve's direction follows increasing parameter numbers.

  3. 03

    When dx/dt is nonzero, dy/dx equals (dy/dt)/(dx/dt).

  4. 04

    Parametric arc length uses the square root of (dx/dt)²+(dy/dt)².

  5. 05

    Polar coordinates describe a point by radius r and angle θ.

  6. 06

    Negative polar radius places a point in the opposite angular direction.

  7. 07

    Polar area from α to β equals one half times the integral of r² dθ.

  8. 08

    Intersections in polar form may occur with different coordinate pairs for the same point.

PART OF THE CALCULUS II GUIDE

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COMMON MIX-UPS

Two ideas worth correcting now

NOT QUITE

dy/dx equals dy/dt in a parametric curve

→
USE THIS INSTEAD

The slope divides dy/dt by dx/dt.

NOT QUITE

One polar point has one coordinate pair

→
USE THIS INSTEAD

Infinitely many polar pairs can name the same point.

ACTIVE RECALL

Flashcards

Answer before opening each card. The effort to retrieve is part of the learning.

1What gives parametric slope?Show answer ＋

(dy/dt)/(dx/dt), when dx/dt is nonzero.

2How is a horizontal parametric tangent found?Show answer ＋

dy/dt = 0 while dx/dt is nonzero.

3How is a vertical parametric tangent found?Show answer ＋

dx/dt = 0 while dy/dt is nonzero.

4How do polar and Cartesian coordinates connect?Show answer ＋

x = r cos θ and y = r sin θ.

5What does a negative r do?Show answer ＋

It points |r| units opposite θ.

6What is polar area?Show answer ＋

One half of ∫r² dθ over the traced interval.

7Why must the tracing interval be checked?Show answer ＋

A long interval may repeat part or all of the curve.

8What is parametric arc length?Show answer ＋

∫√((dx/dt)²+(dy/dt)²) dt.

PRACTICE QUESTIONS

Explain it in your own words

Use the answer as a check after you have written or spoken your response.

01For x=t² and y=t³, what is dy/dx away from t=0?

(3t²)/(2t) = 3t/2.

02How do you locate a polar curve's origin crossings?

Solve r(θ)=0 and include any equivalent coordinate descriptions.

03Why can two polar pairs name one point?

Angles repeat by 2π, and a negative radius reverses direction by π.

04How is area between two polar curves found?

Find one half of the accumulated outer radius squared minus inner radius squared over intervals where their order is known.

05How do you prevent tracing a loop twice?

Find a parameter interval that covers the loop once by checking zeros, symmetry, and direction.

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