Double Integrals in Polar Coordinates · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Round regions want radius and angle

Mathematics · Calculus & Analysis · ages 20-21
Name ______________________   Date ____________

  1. A polar cell spans 2 <= r <= 4 and has angular width 1/2 radian. Its exact area is 3 square units.

    Circle one:   True   False

  2. Before multiplying by the area element, which expression is the polar form of x + y?

    • r (cos theta - sin theta)
    • r squared (cos theta + sin theta)
    • r (cos theta + sin theta)
    • r (cos theta + cos theta)

  3. A sector has vertex at the origin, radius 6 and angular interval pi/6 <= theta <= pi/3. Its area is k pi. What is k?

    Answer: ______________

  4. An annulus centred at the origin has inner radius 1 and outer radius 3. Its area is k pi. What is k?

    Answer: ______________

  5. With x = r cos theta and y = r sin theta, what is the polar area element dA?

    • dr dtheta
    • r dr dtheta
    • r squared dr dtheta

  6. Which polar limits describe the unit disc x squared plus y squared <= 1?

    • 0 <= r <= 2; 0 <= theta <= 2 pi
    • 1 <= r <= 2; 0 <= theta <= 2 pi
    • 0 <= r <= 1; 0 <= theta <= 2 pi

  7. Evaluate the single integral from r = 0 to 2 of r dr.

    Answer: ______________

  8. A point has x = 1 and y = 1. Its polar radius squared, r squared, is a whole number. What is it?

    Answer: ______________

  9. Mia says the extra r appears because polar grid cells get wider as r grows. Is Mia right?

    • Yes, arc length is r times the fixed angular width.
    • No, arc length is the fixed angular width divided by r.
    • No, the radial width must be multiplied by r instead.

  10. For volume under z = x squared + y squared over the unit disc, what does the height function z become in polar form, before multiplying by the area element?

    • r
    • r squared
    • theta squared
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Round regions want radius and angle W1-mt_93e-KSzQMT-s1

  1. True · Subtract the sector areas: ((4 squared - 2 squared) / 2) times 1/2 = 3.
  2. r (cos theta + sin theta) · Substitute x = r cos theta and y = r sin theta, then factor out r. The result is r (cos theta + sin theta).
  3. 3 · The angular width is pi/6. The radial integral is 36/2 = 18, so the area is 18 times pi/6 = 3 pi.
  4. 8 · The area is pi (3 squared - 1 squared) = 8 pi, so k = 8.
  5. r dr dtheta · Changing variables scales area by r, so dA equals r dr dtheta.
  6. 0 <= r <= 1; 0 <= theta <= 2 pi · Radius runs 0 to 1 and the angle sweeps the full circle.
  7. 2 · An antiderivative is r squared over 2, giving 4 over 2.
  8. 2 · r squared equals 1 plus 1, which is 2.
  9. Yes, arc length is r times the fixed angular width. · Wider cells at larger r mean area scales with r.
  10. r squared · The height becomes r squared. Multiplying it by dA = r dr dtheta gives r cubed dr dtheta for the volume integral.
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