Work the Following on Notebook Paper

CALCULUS BC

WORKSHEET 1 ON POLAR

Work the following on notebook paper.

Convert the following equations to polar form.

1. y = 42. 3.

______

Convert the following equations to rectangular form.

4. 5. 6.

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For the following, find for the given value of .

7. 9.

8. 10.

______

11. Find the points of horizontal and vertical tangency for . Give your answers

in polar form, .

______

Use your calculator on problem 12.

12, Given the polar curve

(a) Sketch the graph of the curve.

(b) Find the angle that corresponds to the point(s) on the curve where .

(d) For what values of , is positive? What does this say about r?

What does it say about the curve?

CALCULUS BC

WORKSHEET 2 ON POLAR

For each of the following, sketch a graph, shade the region, and find the area.

Do not use your calculator.

1. one petal of

2. one petal of

3. interior of

4. interior of

5. interior of

6. inner loop of

7. between the loops of

CALCULUS BC

WORKSHEET 3 ON POLAR

For each of the following, sketch a graph, shade the region, and find the area.

Do not use your calculator.

1. inside and outside

2. common interior of

3. inside and outside

4. common interior of

5. common interior of

CALCULUS BC

WORKSHEET 4 ON POLAR

Work the following on notebook paper. Do not use your calculator except on problem 10.

1. Find the slope of the curve .

2. Find the equation of the tangent line to the curve .

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On problems 3 – 5, set up an integral to find the area of the shaded region. Do not evaluate.

3. 4. 5.

______

6. Sketch the polar region described by the following integral expression for area:

______

7. (a) In polar coordinates, write equations for the line x = 1 and the circle of radius 2 centered

at the origin.

(b) Write the integral in polar coordinates representing the area of the region to the right of

x = 1 and inside the circle.

______

8. (a) Sketch the bounded region inside the lemniscate and outside the

circle .

(b) Compute the area of the region described in part (a).

______

9. Find the area between the two spirals and for .

______

Use your calculator on problem 10.

10. Given the polar curve

(a) Sketch the graph of the curve.

(b) Find the angle that corresponds to the point(s) on the curve where .

CALCULUS BC

WORKSHEET 5 ON POLAR

Work the following on notebook paper. You may use your calculator on problems 2 – 5.

On problems 1 – 2, sketch a graph, shade the region, and find the area.

1. inside and outside

2. inside and outside

3. The figure shows the graphs of the line and

the curve C given by . Let S be the region

in the first quadrant bounded by the two graphs and the

x-axis. The line and the curve intersect at point P.

(a) Find the coordinates of P.

(b) Set up and evaluate an integral expression with respect

to x that gives the area of S.

(d) Use the polar equation found in (c) to set up and evaluate

an integral expression with respect to the polar angle

that gives the area of S.

4. A curve is drawn in the xy-plane and is described by the equation in polar coordinates

for , where r is measured in meters and is measured in

radians.

(a) Find the area bounded by the curve and the y-axis.

(b) Find the angle that corresponds to the point on the curve with y-coordinate .

What does it say about the curve?

(d) Find the value of on the interval that corresponds to the point on the curve

with the greatest distance from the origin. What is the greatest distance? Justify your

answer.

5. (From Calculus, 3rd edition, by Finney, Demana, Waits, Kennedy)

A region R in the xy-plane is bounded below by the x-axis and above by the polar curve

defined by

(a) Find the area of R by evaluating an integral in polar coordinates.

(b) The curve resembles an arch of the parabola . Convert the polar equation

to rectangular coordinates, and prove that the curves are the same.

Answers to Worksheet 1 on Polar

1. 10.

2. 11. Horiz:

3. r = 5

4. x = 3 Vert.:

5. 12. (b) 2.786

6. (c) 0.661 and 2.223

7. 0 (d) .

8. – 1r is increasing. The curve is getting farther

9. from the origin. ______

Answers to Worksheet 2 on Polar

1. 3. 5. 47.

2. 4. 6.

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Answers to Worksheet 3 on Polar

1. 3. 5.

2. 4.

______

Answers to Worksheet 4 on Polar

1. 6. 1 petal of 9.

2. 7. (a) 10. (a) graph

3. (b) (b) 1.839, 4.295

4. (c) (c) 0.921, 2.563

5. 8. (a) graph

(b)

Answers to Worksheet 5 on Polar

1. 5. (a) 10.667

2. (b)

3. (a) (1.2, 0.8)

(b) 0.927

(c)

(d) 0.927 (c) 10.667

4. (a) 19.675

(b) 3.485

(d) r

4.302 5.245

4.712 The greatest distance is 5.245 when = 4.302.