Worksheet: Double Integrals over Rectangular Regions

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In this worksheet, we will practice computing double integrals over rectangles by using a sequence of 1-variable integrals.

Q1:

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

Q2:

Evaluate the double integral 𝑥 ( 𝑥 𝑦 + 𝑥 ) 𝑥 𝑦 2 − 1 1 − 1 s i n d d rounding to three decimals.

Q3:

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

Q4:

Evaluate the double integral ( 𝑥 + 2 ) 𝑥 𝑦 2 0 1 0 d d .

Q5:

Evaluate the double integral 𝑥 ( 𝑥 + 𝑦 ) 𝑥 𝑦 1 0 2 0 d d .

  • A10
  • B 5 3
  • C12
  • D 1 1 3
  • E8

Q6:

Evaluate the double integral ( 1 − 𝑦 ) 𝑥 𝑥 𝑦 1 0 2 1 2 d d .

  • A0
  • B 7 2
  • C 7 3
  • D 7 6
  • E 1 6

Q7:

Evaluate the double integral 𝑥 𝑦 𝑦 𝑥 𝑥 𝑦 𝜋 2 0 1 0 2 c o s d d .

  • A0
  • B2
  • C 1 3
  • D 1 2
  • E − 1 2

Q8:

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

Q9:

Find the volume under the surface 𝑧 = 4 𝑥 𝑦 over the rectangle 𝑅 = [ 0 , 1 ] × [ 0 , 1 ] .

Q10:

Find the volume under the surface 𝑧 = 𝑥 + 𝑦 over the rectangle 𝑅 = [ 0 , 1 ] × [ 0 , 1 ] .

  • A 1 3
  • B1
  • C2
  • D 7 1 2
  • E 1 1 2

Q11:

Find the volume under the surface 𝑧 = 𝑥 + 𝑥 𝑦 + 𝑦 4 3 of the rectangular prism with base 𝑅 = [ 1 , 2 ] × [ 0 , 2 ] .

  • A106
  • B80
  • C97
  • D 9 7 5
  • E 1 4 7 5

Q12:

Find the volume under the surface over the rectangle .

  • A
  • B
  • C
  • D
  • E

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