Question

Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

\(0 \le r < 2,\;\;\;{\rm{\pi }} \le {\rm{\theta }} \le \frac{{{\rm{3\pi }}}}{{\rm{2}}}\)

Short answer

Given: \(0 \le r < 2,\;\;\;{\rm{\pi }} \le {\rm{\theta }} \le \frac{{{\rm{3\pi }}}}{{\rm{2}}}\)

Remember that \({\rm{r = a}}\)is a circle with a radius \({\rm{a}}\)cantered at the pole and \({\rm{\theta = b}}\)is a line that passes through the pole and has an angle\({\rm{\theta }}\) with the horizontal in polar coordinates.

The dashed horizontal line is\({\rm{\theta = \pi }}\). (only the portion bordering the region is shown). Because the line itself is included in the region, as shown by the "... or equal" inequality, draw it with a solid line.

The vertical line\({\rm{\theta = }}\frac{{{\rm{3\pi }}}}{{\rm{2}}}\) is identical to the first, except it is solid.

A disc with radius\(0 \le r < 2\). Because\({\rm{r = 2}}\) is not included, this must be rendered with a dashed line. The region is the area of the disc that lies between the two axes.

Step-by-step solution

Definition.

Given: \(0 \le r < 2,\;\;\;{\rm{\pi }} \le {\rm{\theta }} \le \frac{{{\rm{3\pi }}}}{{\rm{2}}}\)

Remember that \({\rm{r = a}}\)is a circle with a radius \({\rm{a}}\)cantered at the pole and \({\rm{\theta = b}}\)is a line that passes through the pole and has an angle\({\rm{\theta }}\) with the horizontal in polar coordinates.

The dashed horizontal line is\({\rm{\theta = \pi }}\). (only the portion bordering the region is shown). Because the line itself is included in the region, as shown by the "... or equal" inequality, draw it with a solid line.

The vertical line\({\rm{\theta = }}\frac{{{\rm{3\pi }}}}{{\rm{2}}}\) is identical to the first, except it is solid.

A disc with radius\(0 \le r < 2\). Because\({\rm{r = 2}}\) is not included, this must be rendered with a dashed line. The region is the area of the disc that lies between the two axes.

Sketch the region.