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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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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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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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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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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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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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