AP®MathsCollege BoardCalculus BCExam QuestionsUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesPolar Coordinates

Polar Coordinates (College Board AP® Calculus BC): Exam Questions

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Polar Coordinates

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1
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2 marks

Let R be the region bounded by the graph of the polar curve r equals theta space cos open parentheses 2 theta close parentheses for 0 less or equal than theta less or equal than pi over 4, as shown in the figure below.

Graph with x and y axes featuring a loop-shaped curve starting at the origin, labelled with "R" inside the loop, on a white background.

Find the area of R.

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2
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2 marks

The graph of the polar curve r equals negative 2 plus 4 space cos space theta for 0 less or equal than theta less or equal than pi over 2 is shown below. The region in the first quadrant bounded by the curve and the x-axis is shaded.

Graph with a shaded area under a curve between x=0 and x=2, with axes labelled x and y. The curve extends down to y=-2.

Write an integral expression for the area of the shaded region.

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3
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2 marks

A curve in the x y-plane is described by the equation r equals theta plus cos open parentheses 2 theta close parentheses in polar coordinates.

For pi over 12 less than theta less than fraction numerator 5 pi over denominator 12 end fraction, fraction numerator d r over denominator d theta end fraction is negative. What does this fact say about r? What does this fact say about the curve?

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4
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2 marks

A curve is given in polar coordinates by the equation r equals 2 theta plus sin space theta for 0 less or equal than theta less or equal than pi.

Find the angle theta that corresponds to the point on the curve with x-coordinate negative 4.

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5
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3 marks

The graphs of the polar curves r equals 5 and r equals 4 plus 2 space sin space theta are shown below. The curves intersect at theta equals pi over 6 and theta equals fraction numerator 5 pi over denominator 6 end fraction. The region shaded is inside the graph of r equals 4 plus 2 space sin space theta and also outside the graph of r equals 5.

Graph showing two polar curves, both with roughly circular shapes around the origin. One is slightly higher than the other. The area enclosed between the two curves at the very top is shaded.

Write an expression involving an integral for the area of the shaded region.

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1
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3 marks

The polar curves r equals 4 and r equals 3 minus sin open parentheses 3 theta close parentheses are shown in the graph below for 0 less or equal than theta less or equal than pi. Let R be the shaded region inside the graph of r equals 4 and inside the graph of r equals 3 minus sin open parentheses 3 theta close parentheses, as shown.

The graph features a shaded grey area labelled R under part of a semi-circular curve and part of an inner curve. The x and y axes intersect at point O.

Find the area of R.

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2
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2 marks

For the polar curve r equals 4 theta plus 2 cos open parentheses 4 theta close parentheses, find the value of fraction numerator d x over denominator d theta end fraction at theta equals pi over 12.

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3
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2 marks

A particle moves along a curve with polar equation r equals 4 sin space theta plus 2 e to the power of negative theta end exponent so that fraction numerator d theta over denominator d t end fraction equals 2 pi for all times t greater or equal than 0. Find the value of fraction numerator d r over denominator d t end fraction at the point when theta equals 1.

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4
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3 marks

The graphs of the polar curves r equals 2 and r equals 3 minus 2 sin space theta are shown below. The curves intersect when theta equals pi over 6 and theta equals fraction numerator 5 pi over denominator 6 end fraction.

Graph with a heart-shaped curve and a circle, both going around the origin, with the circle slightly higher. The region enclosed between them is shaded.

Find the area of the shaded region.

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5
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3 marks

Find the slope of the line tangent to the polar curve r equals 5 minus 4 cos space theta at theta equals pi over 2.

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1
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3 marks

The graph of the polar curve r equals 4 theta squared sin open parentheses 3 theta close parentheses for 0 less or equal than theta less or equal than pi over 3 is shown below.

Graph of a polar curve resembling a loop in the first quadrant, with axes labelled x and y intersecting at the point 0.

There is a straight line through the origin with positive slope m that divides the area of the region shown into two regions with equal areas.

Write, but do not solve, an equation involving one or more integrals whose solution gives the value of m.

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2
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2 marks

The shaded region shown below is bounded by the graph of the polar curve r equals e to the power of negative theta over 2 end exponent for 0 less or equal than theta less or equal than 2 pi.

A graph of the polar curve r = e to the power of minus theta over 2, showing a spiral inwards to the origin and the area between theta = 0 and theta = 2pi shaded.

For k greater than 0, let A open parentheses k close parentheses be the area of the portion of the shaded region that is also inside the circle r equals k space sin space theta. Find limit as k rightwards arrow infinity of A open parentheses k close parentheses.

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3a
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3 marks

The figure below shows the polar curves r equals f open parentheses theta close parentheses equals 1 plus sin space theta and r equals g open parentheses theta close parentheses equals cos open parentheses theta over 2 close parentheses for 0 less than theta less than pi.

A plot showing two polar curves in the first and second quadrants. The outer one looks like part of a circle and the inner one looks like it spirals inwards towards the origin.

For each theta in 0 less than theta less than pi, let h open parentheses theta close parentheses be the distance between the points with polar coordinates open parentheses f open parentheses theta close parentheses comma space theta close parentheses and open parentheses g open parentheses theta close parentheses comma space theta close parentheses. Write an expression for h open parentheses theta close parentheses and find h subscript A, the average value of h open parentheses theta close parentheses over the interval 0 less than theta less than pi.

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3b
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2 marks

Find the value of theta in the interval 0 less than theta less than pi for which h open parentheses theta close parentheses equals h subscript A. Is the function h open parentheses theta close parentheses increasing or decreasing at this value of theta? Give a reason for your answer.

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4
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3 marks

A particle is moving along the polar curve r equals 3 plus 2 space cos space theta so that at time t seconds, theta equals t cubed. Find the position vector of the particle in terms of t and find the velocity vector at time t equals 0.8.

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5a
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2 marks

For the polar curve r equals tan space theta, show that fraction numerator d y over denominator d theta end fraction equals sin space theta open parentheses 2 plus tan squared space theta close parentheses.

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5b
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3 marks

Show that fraction numerator d squared y over denominator d x squared end fraction equals fraction numerator a minus b cos squared theta over denominator cos to the power of n theta end fraction where a, b and n are positive integers that you should find.

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6a
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1 mark

Curve C is defined by the polar equation r(\theta) = 2 \text{sin}^{2} \theta for 0 \leq \theta \leq \pi. Curve C and the semicircle r = \frac{1}{2} for 0 \leq \theta \leq \pi are shown in the x y-plane.

Polar graph of curve C (a large loop above the origin peaking near y ≈ 2.1, labelled C) together with the semicircle r = 1/2 (radius 0.5) at the origin, for 0 ≤ θ ≤ π in the xy-plane

(Note: Your calculator should be in radian mode.)

Find the rate of change of r with respect to \theta at the point on curve C where \theta = 1.3. Show the setup for your calculations.

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6b
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3 marks

Find the area of the region that lies inside curve C but outside the graph of the polar equation r = \frac{1}{2}. Show the setup for your calculations.

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6c
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3 marks

It can be shown that \frac{\text{d} x}{\text{d} \theta} = 4 \text{sin} \theta \text{cos}^{2} \theta - 2 \text{sin}^{3} \theta for curve C. For 0 \leq \theta \leq \frac{\pi}{2}, find the value of \theta that corresponds to the point on curve C that is farthest from the y-axis. Justify your answer.

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6d
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2 marks

A particle travels along curve C so that \frac{\text{d} \theta}{\text{d} t} = 15 for all times t. Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where \theta = 1.3. Show the setup for your calculations.

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7a
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2 marks

The polar curves r = f(\theta) = 1 + \text{sin} \theta \text{cos}(2 \theta) and r = g(\theta) = 2 \text{cos} \theta for 0 \leq \theta \leq \frac{\pi}{2} are shown below. Let R be the region in the first quadrant bounded by the curve r = f(\theta) and the x-axis. Let S be the region in the first quadrant bounded by the curve r = f(\theta), the curve r = g(\theta), and the x-axis.

Two polar curves in the first quadrant of the xy-plane with origin O; 1 and 2 marked on the x-axis and 1 on the y-axis. The lower curve r = f(theta) starts at O, rises to about (0.8, 0.7), curves down through about (1.2, 0.4), and meets the x-axis at (1, 0). The upper curve r = g(theta) = 2cos(theta) is a larger arc from O up and over, meeting the x-axis at (2, 0). Region R is enclosed between the lower curve r = f(theta) and the x-axis. Region S is between the two curves, above r = f(theta) and below r = g(theta).

Find the area of R.

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7b
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2 marks

The ray \theta = k, where 0 < k < \frac{\pi}{2}, divides S into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of k.

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7c
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3 marks

For each \theta with 0 \leq \theta \leq \frac{\pi}{2}, let w(\theta) be the distance between the points with polar coordinates left parenthesis f left parenthesis theta right parenthesis comma space theta right parenthesis and left parenthesis g left parenthesis theta right parenthesis comma space theta right parenthesis. Write an expression for w(\theta). Find w_{A}, the average value of w(\theta) over the interval 0 \leq \theta \leq \frac{\pi}{2}.

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7d
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2 marks

Using the information from part (c), find the value of \theta for which w(\theta) = w_{A}. Is the function w(\theta) increasing or decreasing at that value of \theta? Give a reason for your answer.

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8a
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2 marks

Let S be the region bounded by the graph of the polar curve r(\theta) = 3 \sqrt{\theta} \text{sin} \left(\theta^{2}\right) for 0 \leq \theta \leq \sqrt{\pi}, as shown in the figure above.

Polar region S in the xy-plane with origin O. The curve starts at the origin, rises almost vertically (curving slightly to the left at first), then turns up and to the right, crossing the y-axis above O, and reaches a maximum high above the x-axis and to the right of the y-axis. It then comes down and to the left, ending back at the origin. The enclosed region is shaded and labelled S.

Find the area of S.

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8b
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2 marks

What is the average distance from the origin to a point on the polar curve r(\theta) = 3 \sqrt{\theta} \text{sin} \left(\theta^{2}\right) for 0 \leq \theta \leq \sqrt{\pi}?

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8c
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3 marks

There is a line through the origin with positive slope m that divides the region S into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of m.

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8d
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2 marks

For k > 0, let A(k) be the area of the portion of region S that is also inside the circle r = k \text{cos} \theta. Find \lim_{k \to \infty} A(k).

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9a
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3 marks

The graphs of the polar curves r = 4 and r = 3 + 2 \text{cos} \theta are shown in the figure below. The curves intersect at \theta = \frac{\pi}{3} and \theta = \frac{5 \pi}{3}.

Two polar curves in the xy-plane with origin O and 1 marked on each axis. The circle r = 4 is centered at the origin. The curve r = 3 + 2cos(theta) starts on the positive x-axis at x = 5, loops up and to the left crossing the y-axis near y = 3 and reaching the negative x-axis near x = -1, with its lower half symmetric below the x-axis. The region R, lying inside the circle r = 4 and outside this curve, is shaded; it is the crescent on the left between the two curves.

Let R be the shaded region that is inside the graph of r = 4 and also outside the graph of r = 3 + 2 \text{cos} \theta. Write an expression involving an integral for the area of R.

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9b
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3 marks

Find the slope of the line tangent to the graph of r = 3 + 2 \text{cos} \theta at \theta = \frac{\pi}{2}.

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9c
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3 marks

A particle moves along the portion of the curve r = 3 + 2 \text{cos} \theta for 0 < \theta < \frac{\pi}{2}. The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of 3 units per second. Find the rate at which the angle \theta changes with respect to time at the instant when the position of the particle corresponds to \theta = \frac{\pi}{3}. Indicate units of measure.

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