Unit 5 · Lesson 2 of 7

A swept angle builds area from thin sectors.

A cardioid is easier to describe by distance and direction from a pole. Sweep its angle and watch the enclosed region grow.

Change the input. Follow the mathematics.

What you'll be able to do

Set up the model and calculate its swept area.
Connect the equation to the changing visual.
Explain the assumptions and identify a misleading interpretation.

Predict before exploring

What will change if you increase swept angle b? Predict the direction, then explain the response.

A model you can changeMove one control at a time.

Read the model

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

Swept cardioid

Swept angle b3.14 rad

swept area

A=\frac{3(3.14)}{4}+\sin(3.14)+\frac{\sin(6.28)}{8}=2.35619

A thin polar sector has area approximately ½r² dθ. The parameter is angle, and r is distance from the pole.

Assumptions, units, and limits

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

From the situation to an equation

Convert to Cartesian coordinates with $x=r$ cosθ and $y=r$ sinθ when needed. A small sector contributes one half radius squared times angle increment. Integrate only over an interval that counts the intended region once.

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Keep the reasoning visible

x=r\cos\theta,\quad y=r\sin\theta,\quad dA=\frac12r^2d\theta

A thin polar sector has area approximately ½r² dθ. The parameter is angle, and r is distance from the pole.

What makes this model useful?

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

A polar curve can retrace a region. Bounds and radius signs must be interpreted geometrically before applying the area formula.

A worked example

Calculate the starting scenario

Use swept angle b = 3.14 rad.
$A=\frac{3(3.14)}{4}+\sin(3.14)+\frac{\sin(6.28)}{8}=2.35619$
A thin polar sector has area approximately ½r² dθ. The parameter is angle, and r is distance from the pole. The checked quantity is swept area.

Make the idea your own

Calculate. Compare. Explain.

Numeric checks allow 1% rounding error. Values near zero use a small absolute tolerance. Work through the setup before opening a solution.

Calculate and explain

Rebuild the starting calculation

Calculate the swept area using swept angle b 3.14 rad. State the meaning of its sign and units.

Checked quantity: swept area · square units. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Convert to Cartesian coordinates with x=r cosθ and y=r sinθ when needed. A small sector contributes one half radius squared times angle increment. Integrate only over an interval that counts the intended region once.

Explore the visual solution

Read the model

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

Swept cardioid

Swept angle b3.14 rad

swept area

A=\frac{3(3.14)}{4}+\sin(3.14)+\frac{\sin(6.28)}{8}=2.35619

A thin polar sector has area approximately ½r² dθ. The parameter is angle, and r is distance from the pole.

Assumptions, units, and limits

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Calculate and explain

Apply the idea to a new scenario

Sweep the cardioid to angle b=π radians. Find swept area.

Checked quantity: swept area · square units. Follow the definition for finite-cutoff and numerical-model questions.

Hint

Explore the visual solution

Read the model

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

Swept cardioid

Swept angle b3.142 rad

swept area

A=\frac{3(3.14159)}{4}+\sin(3.14159)+\frac{\sin(6.28319)}{8}=2.35619

A thin polar sector has area approximately ½r² dθ. The parameter is angle, and r is distance from the pole.

Assumptions, units, and limits

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The controls explore variations. These written steps retain the original question's inputs.

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Challenge the interpretation

Which explanation survives the model?

Choose the statement you can justify, then use the visual to explain why the others fail.

Test it with the visual

Read the model

r(\theta)=1+\cos\theta,\quad A=\frac12\int_0^b(1+\cos\theta)^2\,d\theta=\frac{3b}{4}+\sin b+\frac{\sin2b}{8}

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

Swept cardioid

Swept angle b3.14 rad

swept area

A=\frac{3(3.14)}{4}+\sin(3.14)+\frac{\sin(6.28)}{8}=2.35619

A thin polar sector has area approximately ½r² dθ. The parameter is angle, and r is distance from the pole.

Assumptions, units, and limits

Cardioid traced once on 0≤θ≤2π; swept region has nonnegative r and no repeated full turns.

The controls specify a new scenario. Curves are calculated from the equations; physical illustrations are schematic.

The reasoning. Convert to Cartesian coordinates with $x=r$ cosθ and $y=r$ sinθ when needed. A small sector contributes one half radius squared times angle increment. Integrate only over an interval that counts the intended region once.

Your learning path

Can you explain the connection?

Mark complete when you can set up the calculation and explain what the visual is showing. Opening a page never completes it automatically.

Completion is saved on this browser. This is separate from account practice progress.