Free Printable The area bounded by a polar curve

1.

The area bounded by the spiral

r=\frac{a\theta}{\pi}

from

\theta=0\ to\ \theta=\pi

and the initial line is

a)

$\frac{1}{2}\pi a^2$

b)

$\frac{1}{3}\pi a^2$

c)

$\pi a^2$

d)

$\frac{1}{6}\pi a^2$

2.

The area bounded by the spiral $r=\frac{a\theta}{\pi}$ from $\theta=0\$ to $\theta=2\pi$ and the initial line is

a)

$\frac{1}{3}\pi a^2$

b)

$\frac{2}{3}\pi a^2$

c)

$\frac{4}{3}\pi a^2$

d)

$\pi a^2$

3.

The area enclosed by one loop of the curve

r^2=\cos3\theta

is

a)

$\frac{1}{6}\pi$

b)

$\frac{1}{3}\pi$

c)

$\frac{1}{6}$

d)

$\frac{1}{3}$

4.

The area of the cardioid

r=2\left(1+\cos\theta\right)

is

a)

$3\pi$

b)

$12\pi$

c)

$2\pi$

d)

$6\pi$

5.

The area of one loop of the curve

r=a\sin4\theta

is

a)

$\frac{1}{2}\pi a^2$

b)

$\frac{1}{8}\pi a^2$

c)

$\frac{1}{4}\pi a^2$

d)

$\frac{1}{16}\pi a^2$

6.

A curve, C, has a polar equation

r=e^{3\theta},\ 0\ \le\theta\le0.5

.
The area of the region bounded by C and the lines

\theta=0\

and

\theta=0.5

, correct to 3 s.f.,

is

a)

2.68

b)

1.67

c)

1.59

d)

4.95

7.

The area bounded by the curve $r=1+\sqrt{2}\cos\theta\$

is

a)

$\frac{3}{2}\pi+\frac{3}{2}$

b)

$3\pi+3$

c)

$\frac{3}{2}\pi+\frac{5}{2}$

d)

$3\pi+5$

8.

A curve C has polar equation $r=2\cos\theta.$ A curve D has polar equation r = 1 . The line $\theta=0$ intersects C at the point A and D at the point P.

The line $\theta=\frac{\pi}{6}$ intersects C at the point B and D at the point Q.

The area of the region bounded by the arcs PQ and AB and the lines $\theta=0$ and $\theta=\frac{\pi}{6}$ is

a)

$\frac{\sqrt{3}}{16}+\frac{7\pi}{24}$

b)

$\frac{1}{48}\left(14\pi+48+3\sqrt{3}\right)$

c)

$\frac{1}{16}\left(2\pi+8+\sqrt{3}\right)$

d)

$\frac{\pi1}{48}\left(15+8\sqrt{3}\right)$

The area bounded by a polar curve

Total questions: 8

Worksheet time: 24mins

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