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Worksheet: Double Integrals over Rectangular Regions

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

Q1:

Evaluate the double integral ο„Έ ο„Έ π‘₯ ( π‘₯ 𝑦 + π‘₯ ) π‘₯ 𝑦 2 βˆ’ 1 1 βˆ’ 1 s i n d d rounding to three decimals.

Q2:

Find the volume under the surface 𝑧 = π‘₯ + π‘₯ 𝑦 + 𝑦 4 3 over the rectangle 𝑅 = [ 1 , 2 ] Γ— [ 0 , 2 ] .

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

Q3:

Evaluate the double integral ο„Έ ο„Έ π‘₯ 𝑦 ο€Ή 𝑦 π‘₯  π‘₯ 𝑦 πœ‹ 2 0 1 0 2 c o s d d .

  • A0
  • B2
  • C 1 3
  • D 1 2
  • E βˆ’ 1 2

Q4:

Evaluate the double integral ο„Έ ο„Έ π‘₯ ( π‘₯ + 𝑦 ) π‘₯ 𝑦 1 0 2 0 d d .

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

Q5:

Evaluate the double integral ο„Έ ο„Έ π‘₯ ( 𝑦 βˆ’ πœ‹ ) π‘₯ 𝑦 πœ‹ 0 0 πœ‹ 2 s i n c o s d d .

Q6:

Evaluate the double integral ο„Έ ο„Έ ( π‘₯ + 2 ) π‘₯ 𝑦 2 0 1 0 d d .

Q7:

Find the volume under the surface 𝑧 = 4 π‘₯ 𝑦 over the rectangle 𝑅 = [ 0 , 1 ] Γ— [ 0 , 1 ] .

Q8:

Find the volume under the surface 𝑧 = π‘₯ + 𝑦   over the rectangle 𝑅 = [ 0 , 1 ] Γ— [ 0 , 1 ] .

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

Q9:

Find the volume under the surface 𝑧 = 𝑒 ( π‘₯ + 𝑦 ) over the rectangle 𝑅 = [ 0 , 1 ] Γ— [ βˆ’ 1 , 1 ] .

  • A βˆ’ 1 + 1 𝑒 βˆ’ ( 𝑒 βˆ’ 1 ) 𝑒
  • B βˆ’ 1 βˆ’ 1 𝑒 + ( 𝑒 βˆ’ 1 ) 𝑒
  • C βˆ’ 1 βˆ’ 1 𝑒 βˆ’ ( 𝑒 βˆ’ 1 ) 𝑒
  • D βˆ’ 1 + 1 𝑒 + ( 𝑒 βˆ’ 1 ) 𝑒
  • E 𝑒 2

Q10:

Evaluate the double integral ο„Έ ο„Έ 1 π‘₯ 𝑦 1 βˆ’ 1 2 βˆ’ 1 d d .

Q11:

Evaluate the double integral ο„Έ ο„Έ ( 1 βˆ’ 𝑦 ) π‘₯ π‘₯ 𝑦 1 0 2 1 2 d d .

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

Q12:

Evaluate the double integral ο„Έ ο„Έ π‘₯ 𝑦 π‘₯ 𝑦 2 0 4 1 d d .

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