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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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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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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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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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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
Sme Calculator3 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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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
Sme Calculator2 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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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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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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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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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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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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3
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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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4a
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For the polar curve r equals tan space theta, show that fraction numerator d x over denominator d theta end fraction equals cos space theta.

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4b
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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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