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# Sketch a circle tangent to the y-axis and compute its area from -π/2 to π/2

Define a circle tangent to the -axis by:

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Sketch the Spiral of Archimedes and compute its area from 0 to 2 π

Define
Spiral of Archimedes: for .

Sketch this graph in polar coordinates and compute the area of the radial set.

The sketch is as follows:

Then, we compute the area,

# Show an equivalence between given sets in Cartesian and polar coordinates

Show that the sets of points satisfying

are equal.

Letting and we plug in to the given Cartesian equation,

# Show an equivalence between given sets in Cartesian and polar coordinates

Show that the sets of points satisfying

are equal.

Letting and we plug in to the given Cartesian equation,

# Show an equivalence between given sets in Cartesian and polar coordinates

Show that the sets of points satisfying

are equal.

Letting and we plug in to the given Cartesian equation,

# Show an equivalence between given sets in Cartesian and polar coordinates

Show that the sets of points satisfying

are equal.

Letting and we plug in to the given Cartesian equation,

where so (hence, we can divide by ) in the second to last line.