Area of r=a between \theta=0 and \theta=\frac{\pi}{3}?

\frac12a^2\cdot\frac{\pi}{3}=\frac{\pi a^2}{6}

Identity for integrating \cos^2\theta?

\cos^2\theta=\frac12(1+\cos2\theta)

Identity for integrating \sin^2\theta?

\sin^2\theta=\frac12(1-\cos2\theta)

Area between two polar curves, r1\ge r2?

\frac12\int\left(r1^2-r2^2\right)d\theta

Condition for a tangent parallel to the initial line?

\frac{dy}{d\theta}=0 where y=r\sin\theta

Condition for a tangent perpendicular to the initial line?

\frac{dx}{d\theta}=0 where x=r\cos\theta

Why can you not use \frac{dr}{d\theta}=0 for horizontal tangents?

It gives greatest and least r, not the direction of the tangent.

Common error in expanding (3+2\cos\theta)^2 for area?

Dropping the cross term 12\cos\theta.

Area enclosed by a polar curveEdexcel A-Level Further Maths: Flashcards

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Formula for the area enclosed by a polar curve?

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Formula for the area enclosed by a polar curve?: $\frac12\int_\alpha^\beta r^2\,d\theta$
What must be true about the angle for the area formula?: It is measured in radians.
Area of $r=a$ between $\theta=0$ and $\theta=\frac{\pi}{3}$ ?: $\frac12a^2\cdot\frac{\pi}{3}=\frac{\pi a^2}{6}$
Identity for integrating $\cos^2\theta$ ?: $\cos^2\theta=\frac12(1+\cos2\theta)$
Identity for integrating $\sin^2\theta$ ?: $\sin^2\theta=\frac12(1-\cos2\theta)$
Area of the cardioid $r=1+\cos\theta$ ?: $\frac{3\pi}{2}$
Area between two polar curves, $r_1\ge r_2$ ?: $\frac12\int\left(r_1^2-r_2^2\right)d\theta$
How do you find the limits where two curves meet?: Solve $r_1=r_2$ for $\theta$ .
Condition for a tangent parallel to the initial line?: $\frac{dy}{d\theta}=0$ where $y=r\sin\theta$
Condition for a tangent perpendicular to the initial line?: $\frac{dx}{d\theta}=0$ where $x=r\cos\theta$
Why can you not use $\frac{dr}{d\theta}=0$ for horizontal tangents?: It gives greatest and least $r$ , not the direction of the tangent.
Area of $r=2\cos\theta$ for $-\frac{\pi}{2}\le\theta\le\frac{\pi}{2}$ ?: $\pi$ (a circle of radius $1$ )
Common error in expanding $(3+2\cos\theta)^2$ for area?: Dropping the cross term $12\cos\theta$ .

Exam questions on Area enclosed by a polar curve

The curve $C$ has polar equation $r=2\theta$ for $0\le\theta\le\pi$ .
Find the exact area of the region bounded by $C$ and the half-lines $\theta=\frac{\pi}{2}$ and $\theta=\pi$ .2 marks
The curve $C$ has polar equation $r=3+2\cos\theta$ for $0\le\theta\le2\pi$ .
Find the exact area of the region bounded by $C$ and the half-lines $\theta=0$ and $\theta=\frac{\pi}{2}$ .2 marks
The curve $C$ has polar equation $r=2(1+\cos\theta)$ for $0\le\theta\le\pi$ .
Find the polar coordinates of the point of $C$ , other than the pole and the point where $\theta=0$ , at which the tangent is perpendicular to the initial line.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).