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Worksheet: Double Integrals on General Regions

Q1:

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

Q2:

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

  • A2
  • B 2 π‘₯ 2
  • C 2 π‘₯
  • D 2 3
  • E1

Q3:

Evaluate the double integral ο„Έ ο„Έ π‘₯ π‘₯ 𝑦 πœ‹ 0 𝑦 0 s i n d d .

  • A πœ‹ 2
  • B 1 βˆ’ πœ‹
  • C 1 βˆ’ πœ‹ 2
  • D πœ‹
  • E0

Q4:

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

  • A πœ‹
  • B πœ‹ 2
  • C 2 πœ‹ 3
  • D πœ‹ 4
  • E 1 3

Q5:

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

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

Q6:

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

Q7:

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

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

Q8:

Find the volume 𝑉 of the solid bounded by the three coordinate planes and the plane π‘₯ + 𝑦 + 𝑧 = 1 .

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

Q9:

Find the volume of the solid 𝑆 bounded by the three coordinate planes, the plane π‘₯ + 𝑦 + 𝑧 = 2 from above, and the plane 𝑧 = π‘₯ + 𝑦 from below.

  • A2
  • B 2 3
  • C 4 3
  • D 1 3
  • E βˆ’ 4 3

Q10:

Evaluate ο„Έ ο„Έ ο€Ό π‘₯ + 𝑦 2  ο€» π‘₯ βˆ’ 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 βˆ’ 2 s i n
  • B s i n 2 2
  • C 1 + 2 2 s i n
  • D 1 βˆ’ 2 2 s i n
  • E 1 + 2 s i n