Worksheet: Double Integrals in Polar Coordinates

In this worksheet, we will practice evaluating a double integral in polar coordinates and converting a double integral from Cartesian coordinates to polar coordinates.

Q1:

Find the volume inside the cone 𝑧 = 𝑥 + 𝑦 , where 0 𝑧 3 .

  • A 1 8 𝜋
  • B 9 2 3 𝜋
  • C 9 2
  • D 3 𝜋
  • E 9 𝜋

Q2:

Find the volume inside the surface 𝑧 = 𝑥 + 𝑦 , where 0 𝑧 4 .

  • A 8 𝜋
  • B 4 𝜋
  • C 6 𝜋
  • D 1 6 𝜋
  • E 3 𝜋

Q3:

Find, in terms of 𝜋 , the volume of the region that lies within both the sphere with equation 𝑥 + 𝑦 + 𝑧 = 4 and the cylinder with equation 𝑥 + 𝑦 = 1 .

  • A 2 3 𝜋
  • B 3 𝜋
  • C 4 𝜋 3 8 3
  • D 2 𝜋 3 8 3
  • E 𝜋 3 8 3

Q4:

Find the volume inside both the sphere 𝑥 + 𝑦 + 𝑧 = 1 and the cone 𝑧 = 𝑥 + 𝑦 .

  • A 2 𝜋 3 1 1 2
  • B 𝜋 6 1 1 2
  • C 𝜋 1 2 7 3 3
  • D 𝜋 9 1 1 2
  • E 𝜋 3 1 1 2

Q5:

Evaluate the integral 𝑒 𝑥 d .

  • A 2 𝜋
  • B 2 𝜋
  • C 2 𝜋
  • D 𝜋
  • E 3 𝜋

Q6:

For 𝜎 > 0 and 𝜇 > 0 , evaluate 1 𝜎 2 𝜋 𝑒 𝑥 . ( ) d

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