Worksheet: Double Integrals over General Regions

In this worksheet, we will practice extending the calculation of double integrals over nonrectangular regions.

Q1:

Evaluate the double integral 𝑥𝑥𝑦sindd.

  • A1𝜋2
  • B1𝜋
  • C𝜋
  • D0
  • E𝜋2

Q2:

Evaluate the double integral 1𝑥𝑦dd.

Q3:

Evaluate the double integral 2𝑦𝑥dd.

  • A1
  • B2
  • C2𝑥
  • D2𝑥
  • E23

Q4:

Evaluate the double integral 𝑥𝑦𝑥𝑦cossindd.

  • A𝜋2
  • B𝜋
  • C13
  • D𝜋4
  • E2𝜋3

Q5:

Evaluate the double integral 𝑒𝑥𝑦dd.

  • A𝑒
  • B2𝑒1
  • C1𝑒
  • D𝑒1
  • E𝑒1

Q6:

Evaluate the double integral 24𝑦𝑥𝑦𝑥dd.

Q7:

Evaluate the double integral 4𝑥𝑦𝑥.lndd

  • A423ln
  • Bln434
  • C824ln
  • D823ln
  • E424ln

Q8:

Find the volume 𝑉 of the solid bounded by the three coordinate planes and the plane 𝑥+𝑦+𝑧=1.

  • A16
  • B76
  • C12
  • D13
  • E23

Q9:

Evaluate 𝑥+𝑦2𝑥𝑦2𝐴sincosd, where 𝑅 is the triangle with the vertices (0,0), (2,0), and (1,1).

  • A1+22sin
  • B12sin
  • Csin22
  • D122sin
  • E1+2sin

Q10:

Find the volume inside the elliptic cylinder 𝑥𝑎+𝑦𝑏=1, where 0𝑧2.

  • A4𝜋(𝑎𝑏)
  • B𝜋(𝑎𝑏)
  • C2𝜋𝑎𝑏
  • D4𝜋𝑎𝑏
  • E𝜋𝑎𝑏

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