Choosing the interval for a polar petal

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

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Before you start

Areas in polar coordinates. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

A polar curve has equation $r=4\sin(3\theta)$ .

Axes for your polar sketch. Negative radii lie on the opposite ray.

Part a

Identify bounds which trace one petal exactly once and justify them.

Part b

Find the area of one petal as a coefficient of π.

Part c

Sketch the full curve and explain whether a full turn retraces it.

Model solution and review criteria

Part a

Use 0 ≤ θ ≤ $\frac{\pi}{3}$ . Consecutive zeros occur at these endpoints; r is nonnegative inside and reaches 4 at θ = $\frac{\pi}{6}$ .

Locate consecutive zeros of r.
Check the trace between them rather than choosing a full turn.

Part b

Area = ½∫₀^( $\frac{\pi}{3}$ ) 16sin²(3θ)dθ. Using sin²u = (1−cos2u)/2 gives $\frac{4}{3}$ π.

Use the polar area factor ½.
Use a correct trigonometric identity and the chosen bounds.

Part c

There are 3 petals. A full turn retraces each petal, so integrating $r^{2}$ through 2π counts the area twice.

Give the correct petal count.
Explain the role of negative radii and repeated tracing.

Compare your own reasoning. No automatic examination marks are awarded.