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 𝑥 𝑥 𝑦 𝜋 0 𝑦 0 s i n d d .

  • A 𝜋 2
  • B 1 𝜋
  • C 1 𝜋 2
  • D 𝜋
  • E0

Q2:

Evaluate the double integral 1 𝑥 𝑦 2 0 𝑦 0 d d .

Q3:

Evaluate the double integral 2 𝑦 𝑥 1 0 𝑥 0 2 d d .

  • A2
  • B 2 𝑥 2
  • C 2 𝑥
  • D 2 3
  • E1

Q4:

Evaluate the double integral 𝑥 𝑦 𝑥 𝑦 𝜋 2 0 𝑦 0 c o s s i n d d .

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

Q5:

Evaluate the double integral 𝑒 𝑥 𝑦 2 0 2 𝑦 0 𝑦 2 d d .

  • A 1 𝑒 4
  • B 2 𝑒 1 4
  • C 𝑒 4
  • D 𝑒 1 4
  • E 𝑒 1 2

Q6:

Evaluate the double integral 2 4 𝑦 𝑥 𝑦 𝑥 1 0 1 𝑥 2 d d .

Q7:

Evaluate the double integral 4 𝑥 𝑦 𝑥 . 2 1 𝑥 0 l n d d

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

Q8:

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

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

Q9:

Evaluate 𝑥 + 𝑦 2 𝑥 𝑦 2 𝐴 𝑅 s i n c o s d , where 𝑅 is the triangle with the vertices ( 0 , 0 ) , ( 2 , 0 ) , and ( 1 , 1 ) .

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

Q10:

Find the volume inside the elliptic cylinder 𝑥 𝑎 + 𝑦 𝑏 = 1 2 2 2 2 , where 0 𝑧 2 .

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

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