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

Round regions want radius and angle

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

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

    Circle one:   True   False

  2. The point (3, 4) in rectangular coordinates has polar radius r. What is r?

    • 5
    • 7
    • 25
    • 1

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

    Answer: ______________

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

    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 with 0 <= theta <= pi
    • 1 <= r <= 2 with 0 <= theta <= 2 pi
    • 0 <= r <= 1 with 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, a cell angular side has length r times angle
    • No, cells shrink as r grows
    • No, the r comes from the integrand

  10. For the volume under z = x squared plus y squared over the unit disc, what does the integrand become in polar form?

    • 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 · Mia is right: a cell of fixed angular width has arc length r times angle, so area grows with r.
  2. 5 · Radius is the hypotenuse: sqrt(9 + 16) = 5.
  3. 2 · An antiderivative is r squared over 2, giving 4/2 = 2.
  4. 2 · r squared = 1 + 1 = 2.
  5. r dr dtheta · Changing variables scales area by r, so dA equals r dr dtheta.
  6. 0 <= r <= 1 with 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, a cell angular side has length r times angle · Wider cells at larger r mean area scales with r.
  10. r squared · x squared plus y squared is r squared, and the area element adds one more r.
Worksheet · LightMySky