Lesson Worksheet: Double Integrals over Rectangular Regions Mathematics

In this worksheet, we will practice evaluating a double integral over a rectangular region by using iterated integrals.

Q1:

What is the value of the double integral ο„Έο„Έ11π‘₯𝑦π‘₯π‘¦οŠ§οŠ¦οŠ§οŠ¦οŠ¨οŠ©dd?

  • A116
  • B11
  • C0
  • D112
  • E1112

Q2:

Fill in the blank: The value of ο„Έο„Έπ‘₯2π‘₯𝑦π‘₯π‘¦οŽ„οŠ±οŽ„οŠ¨οŽ„οŠ±οŠ¨οŽ„οŠ¨sincosdd is .

  • A2πœ‹
  • B2πœ‹οŠ¨
  • Cβˆ’2πœ‹
  • Dβˆ’4πœ‹οŠ¨
  • Eβˆ’2πœ‹οŠ¨

Q3:

Which of the following is the value of ο„Έο„Έ(π‘₯+𝑦)π‘₯π‘¦οŠ§οŠ¦οŠ¦οŠ±οŠ§ο‰dd, where π‘š is a positive integer?

  • A1βˆ’(βˆ’1)(π‘š+1)(π‘š+2)
  • B1βˆ’(βˆ’1)(π‘š+1)(π‘š+2)ο‰οŠ°οŠ§
  • C1(π‘š+1)(π‘š+2)
  • D1+(βˆ’1)π‘š+2ο‰οŠ°οŠ¨
  • E1+(βˆ’1)(π‘š+1)(π‘š+2)

Q4:

Evaluate 𝐼=ο„Έο„Έπ‘₯(𝑦)𝐴cosd, where 𝑅=[0,3]×0,πœ‹2.

  • A12
  • B92
  • Cβˆ’12
  • Dβˆ’92
  • E0

Q5:

Find the value of the double integral 𝑒+𝑦1+𝑦𝑦π‘₯ln()οŠ¦οŠ§οŠ¦ο—οŠ¨dd, rounded to the nearest two decimal places.

Q6:

Evaluate 𝑓(π‘₯,𝑦)𝑦π‘₯dd over the rectangle 𝑅=𝑒,π‘’οΓ—ο“πœ‹6,πœ‹3, where 𝑓(π‘₯,𝑦)=(𝑦)π‘₯tan. Estimate your answer to the nearest two decimal places.

Q7:

For the function 𝑓(π‘₯,𝑦)=π‘₯π‘₯βˆšπ‘¦cossin, find 𝑓(π‘₯,𝑦)𝐴d, where 𝑅=5πœ‹2,4πœ‹ο Γ—[4,9].

Q8:

Find the volume of a solid bounded from above by the function 𝑓(π‘₯,𝑦)=π‘₯+π‘¦βˆ’5 and bounded from below by the rectangle 𝑅={(π‘₯,𝑦)βˆ£βˆ’1≀π‘₯≀0,2≀𝑦≀3}.

  • A53
  • B233
  • C23
  • D113
  • E173

Q9:

Evaluate the double integral of the function 𝐹(π‘₯,𝑦)=144βˆ’4π‘₯βˆ’π‘¦οŠ¨οŠ¨ over the square 𝐷 shown in the figure.

Q10:

If 𝐼=𝑒+𝑦𝑒π‘₯π‘¦οŠ§οŠ±οŠ§οŠ±οŠ§οŠ±οŠ¨ο—ο˜οŽ‘dd, find the value of 𝐼.

  • A4ο€Ό1π‘’βˆ’1π‘’οˆοŠ©
  • B2ο€Ό1π‘’βˆ’1π‘’οˆοŠ©
  • C2ο€Ό1π‘’βˆ’1π‘’οˆοŠ¨
  • D8ο€Ό1π‘’βˆ’1π‘’οˆοŠ¨
  • E4ο€Ό1π‘’βˆ’1π‘’οˆοŠ¨

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