Question

3. (8 points) Find the area of the region enclosed by the positive x-axis and the spiral
r=4θ/3, 0≤θ≤2 π i.e. the spiral with vector equation c(t )=< 4θ/3⋅cosθ , 4θ/3⋅sin θ > . Note
that this region looks like a snail shell.

Answer 1

In plane polar coordinates, the element of area dA is given by
\[ r \ dr \ d\theta \]

In this problem, you therefore want to evaluate that over the appropriate range of r and theta

\[ \int_{\theta = 0}^{2\pi} \int_{r=0}^{4\theta/3} r \ dr \ d\theta \]