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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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A polar curve has the equation r equals theta squared plus sin open parentheses theta close parentheses. The distance from the origin to the point on the curve at which theta equals pi over 6 is

  • fraction numerator pi squared plus 3 over denominator 6 end fraction

  • fraction numerator pi squared minus 18 square root of 3 over denominator 36 end fraction

  • fraction numerator pi squared plus 18 over denominator 36 end fraction

  • fraction numerator pi squared plus 18 square root of 3 over denominator 36 end fraction

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2
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The curve x squared plus y squared equals 16 has the polar equation

  • r equals 16

  • r equals 4

  • r equals 2

  • r equals negative 16

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3
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Which of the following represents the area of the region enclosed by the loop of the polar curve r equals sin open parentheses 4 theta close parentheses from 0 less or equal than theta less or equal than pi over 4, shown below?

Graph of the polar curve r = sin(4 \theta) creating a closed loop shape, with the x-axis and y-axis labelled.

  • integral subscript 0 superscript pi over 4 end superscript square root of sin open parentheses 4 theta close parentheses end root space d theta

  • integral subscript 0 superscript pi over 4 end superscript square root of 1 plus 16 cos squared open parentheses 4 theta close parentheses end root space d theta

  • integral subscript 0 superscript pi over 4 end superscript sin open parentheses 4 theta close parentheses space d theta

  • 1 half integral subscript 0 superscript pi over 4 end superscript sin squared open parentheses 4 theta close parentheses space d theta

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4
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Which of the following is not equal to the shaded area in the first quadrant that is inside the polar curve r equals 2 and outside the polar curve r equals 1 plus cos space theta, shown below?

Graph in polar coordinates showing shaded region between curves \( r = 2 \) and \( r = 1 + \cos \theta \) on the right side from the origin.

  • 1 half integral subscript 0 superscript pi over 2 end superscript open parentheses 2 minus open parentheses 1 plus cos space theta close parentheses close parentheses squared space d theta

  • 1 half integral subscript 0 superscript pi over 2 end superscript 4 space d theta minus 1 half integral subscript 0 superscript pi over 2 end superscript open parentheses 1 plus cos space theta close parentheses squared space d theta

  • pi minus 1 half integral subscript 0 superscript pi over 2 end superscript open parentheses 1 plus cos space theta close parentheses squared space d theta

  • 1 half integral subscript 0 superscript pi over 2 end superscript open parentheses 4 minus open parentheses 1 plus cos space theta close parentheses squared close parentheses d theta

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5
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The value of fraction numerator d r over denominator d theta end fraction at theta equals pi over 2 on the polar curve r equals 3 cos space theta minus 2 sin space theta shows that, at that point,

  • the curve is at its minimum distance from the origin.

  • the curve is at its maximum distance from the origin.

  • the points on the curve are moving closer to the origin as theta increases.

  • the points on the curve are moving further away from the origin as theta increases.

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1
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The area of the shaded region shown on the graph of the polar curve r equals 3 plus 6 sin space theta is

Graph of a polar curve with equation r = 3 + 6sin(θ), showing a larger outer loop and a shaded smaller inner loop centred on the origin.

  • 0

  • 2.055

  • 4.892

  • 9.783

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2
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The area of the closed region bounded by the polar curve r equals square root of 2 plus sin space theta end root is

  • table row blank blank cell integral subscript 0 superscript 2 pi end superscript open parentheses 1 plus 1 half sin space theta close parentheses space d theta end cell end table

  • 2 integral subscript 0 superscript pi table row blank blank cell open parentheses 1 plus 1 half sin space theta close parentheses end cell end table table row blank blank space end table table row blank blank d end table table row blank blank theta end table

  • table row blank blank cell integral subscript 0 superscript 2 pi end superscript 1 half square root of 2 plus sin space theta end root space d theta end cell end table

  • table row blank blank cell integral subscript 0 superscript pi square root of 2 plus sin space theta end root space d theta end cell end table

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3
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Which of the following represents the graph of the polar curve r equals 3 csc space theta?

  • Graph showing a vertical line intersecting the x-axis at 3 on a Cartesian plane with labelled axes x and y.
  • Graph showing a horizontal line at y equals 3, intersecting the y-axis; x and y axes are labelled with arrows pointing right and up.
  • Graph showing a circle centred at origin with radius 3, intersecting the x-axis at 3, on a Cartesian plane with x and y axes labeled.
  • Graph showing a circle passing through the origin, intersecting the y-axis at 3, centred vertically along the y-axis. Axes are labelled x and y with arrows.

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4
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Which of the following is equal to the area of the region inside the polar curve r equals 3 space sin space theta and outside the polar curve r equals space sin space theta for 0 less or equal than theta less or equal than pi?

  • integral subscript 0 superscript pi 2 sin squared theta space d theta

  • integral subscript 0 superscript pi 4 sin squared theta space d theta

  • integral subscript 0 superscript pi 5 sin squared theta space d theta

  • integral subscript 0 superscript pi 8 sin squared theta space d theta

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5
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The value of fraction numerator d y over denominator d theta end fraction at theta equals pi over 4on the polar curve r equals 4 plus 3 cos space theta is

  • negative 3 minus 2 square root of 2

  • negative fraction numerator 3 square root of 2 over denominator 2 end fraction

  • 2 square root of 2

  • 3 over 2 plus 2 square root of 2

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1
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An expression for the slope of a spiral defined by the polar curve r equals theta, where theta greater than 0, in terms of theta is

  • 1

  • sec squared theta

  • negative cot space theta

  • fraction numerator sin space theta plus theta cos space theta over denominator cos space theta minus theta sin space theta end fraction

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2
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The shaded area shown below, between the two polar curves r equals 4 plus 3 sin space theta and r equals 3 plus sin space theta, has an area of

Plot showing two polar curves, both circle-like but one larger and with a flat bottom. A region bounded by the two curves is shaded.

  • 22.245

  • 27.508

  • 56.802

  • 66.740

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3
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A particle moves along the polar curve r equals 2 plus cos space theta with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses, measured in centimeters, where t is time, measured in seconds. If the motion of the particle is such that the angle theta always increases at a constant rate of 3 radians per second, then the rate at which the distance r decreases at the point where theta equals pi over 6 is

  • 1 over 6 centimeters per second

  • 1.5 centimeters per second

  • 3 centimeters per second

  • 6 plus fraction numerator 3 square root of 3 over denominator 2 end fraction centimeters per second

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4
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A particle moves along the polar curve r equals 2 plus theta squared with position open parentheses x open parentheses t close parentheses comma space y open parentheses t close parentheses close parentheses at time t greater or equal than 0. If fraction numerator d x over denominator d t end fraction equals 20 at the point when theta equals fraction numerator 5 pi over denominator 8 end fraction, then fraction numerator d theta over denominator d t end fraction at this point is

  • 27.747

  • 14.416

  • negative 2.893

  • negative 138.248

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5
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For the circle defined by the polar equation r equals a where a is a positive constant and 0 less or equal than theta less than 2 pi, an expression for fraction numerator d squared y over denominator d x squared end fraction in terms of theta is

  • negative 1 over a csc cubed space theta

  • csc squared space theta

  • 1 over a csc space theta space sec squared space theta

  • negative sec squared space theta

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