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# Convert complex number in polar form to the form a + bi

Convert each of the following complex numbers given in polar form to the form .

1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .

1. Using the definition of the complex exponential ( ) we have 2. Again, using the definition of the complex exponential we have, 3. We have 4. We have 5. We have 6. We have 7. We have 8. We have # Sketch a “limacon” and compute its area from 0 to 2 π

Define a limacon 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 a “cardioid” and compute its area from 0 to 2 π

Define a cardioid 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, where we know from this exercise (Section 2.11, Exercise #7).

# Sketch a “four-leaf clover” and compute its area from 0 to 2 π

Define a four-leaf clover 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, where for the final step we used the previous exercise (Section 2.11, Exercise #12).

# Sketch a “lazy eight” and compute its area from 0 to 2π

Define a lazy eight 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 a “four-leaved rose” and compute its area from 0 to 2π

Define a four-leaved rose 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 a “rose petal” and compute its area from 0 to π/2

Define a rose petal 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 two circles tangent to the x-axis and compute their area from 0 to 2π

Define two circles 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 a circle tangent to the x-axis and compute its area from 0 to π

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 two circles tangent to the y-axis and compute their area from 0 to 2 π

Define two circles 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, 