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.

  • A 1 𝜋 2
  • B 1 𝜋
  • C 𝜋
  • D0
  • E 𝜋 2

Q2:

Evaluate the double integral 1𝑥𝑦dd.

Q3:

Evaluate the double integral 2𝑦𝑥dd.

  • A1
  • B2
  • C 2 𝑥
  • D 2 𝑥
  • E 2 3

Q4:

Evaluate the double integral 𝑥𝑦𝑥𝑦cossindd.

  • A 𝜋 2
  • B 𝜋
  • C 1 3
  • D 𝜋 4
  • E 2 𝜋 3

Q5:

Evaluate the double integral 𝑒𝑥𝑦dd.

  • A 𝑒
  • B 2 𝑒 1
  • C 1 𝑒
  • D 𝑒 1
  • E 𝑒 1

Q6:

Evaluate the double integral 24𝑦𝑥𝑦𝑥dd.

Q7:

Evaluate the double integral 4𝑥𝑦𝑥.lndd

  • A 4 2 3 l n
  • B l n 4 3 4
  • C 8 2 4 l n
  • D 8 2 3 l n
  • E 4 2 4 l n

Q8:

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

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

Q9:

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

  • A 1 + 2 2 s i n
  • B 1 2 s i n
  • C s i n 2 2
  • D 1 2 2 s i n
  • E 1 + 2 s i n

Q10:

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

  • A 4 𝜋 ( 𝑎 𝑏 )
  • B 𝜋 ( 𝑎 𝑏 )
  • C 2 𝜋 𝑎 𝑏
  • D 4 𝜋 𝑎 𝑏
  • E 𝜋 𝑎 𝑏

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