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 0𝑧3.

  • A92
  • B923𝜋
  • C3𝜋
  • D9𝜋
  • E18𝜋

Q2:

Find the volume inside the surface 𝑧=𝑥+𝑦, where 0𝑧4.

  • A3𝜋
  • B4𝜋
  • C6𝜋
  • D16𝜋
  • E8𝜋

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.

  • A2𝜋383
  • B3𝜋
  • C𝜋383
  • D4𝜋383
  • E23𝜋

Q4:

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

  • A𝜋3112
  • B2𝜋3112
  • C𝜋9112
  • D𝜋6112
  • E𝜋12733

Q5:

Evaluate the integral 𝑒𝑥d.

  • A2𝜋
  • B3𝜋
  • C2𝜋
  • D𝜋
  • E2𝜋

Q6:

For 𝜎>0 and 𝜇>0, evaluate 1𝜎2𝜋𝑒𝑥.()d

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