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 11𝑥𝑦𝑥𝑦dd?

  • A116
  • B11
  • C0
  • D112
  • E1112

Q2:

Fill in the blank: The value of 𝑥2𝑥𝑦𝑥𝑦sincosdd is .

  • A2𝜋
  • B2𝜋
  • C2𝜋
  • D4𝜋
  • E2𝜋

Q3:

Which of the following is the value of (𝑥+𝑦)𝑥𝑦dd, where 𝑚 is a positive integer?

  • A1(1)(𝑚+1)(𝑚+2)
  • B1(1)(𝑚+1)(𝑚+2)
  • C1(𝑚+1)(𝑚+2)
  • D1+(1)𝑚+2
  • E1+(1)(𝑚+1)(𝑚+2)

Q4:

Evaluate 𝐼=𝑥(𝑦)𝐴cosd, where 𝑅=[0,3]×0,𝜋2.

  • A12
  • B92
  • C12
  • D92
  • E0

Q5:

Find the value of the double integral 𝑒+𝑦1+𝑦𝑦𝑥ln()dd, rounded to the nearest two decimal places.

Q6:

Evaluate 𝑓(𝑥,𝑦)𝑦𝑥dd over the rectangle 𝑅=𝑒,𝑒×𝜋6,𝜋3, where 𝑓(𝑥,𝑦)=(𝑦)𝑥tan. Estimate your answer to the nearest two decimal places.

Q7:

For the function 𝑓(𝑥,𝑦)=𝑥𝑥𝑦cossin, find 𝑓(𝑥,𝑦)𝐴d, where 𝑅=5𝜋2,4𝜋×[4,9].

Q8:

Find the volume of a solid bounded from above by the function 𝑓(𝑥,𝑦)=𝑥+𝑦5 and bounded from below by the rectangle 𝑅={(𝑥,𝑦)1𝑥0,2𝑦3}.

  • A53
  • B233
  • C23
  • D113
  • E173

Q9:

Evaluate the double integral of the function 𝐹(𝑥,𝑦)=1444𝑥𝑦 over the square 𝐷 shown in the figure.

Q10:

If 𝐼=𝑒+𝑦𝑒𝑥𝑦dd, find the value of 𝐼.

  • A41𝑒1𝑒
  • B21𝑒1𝑒
  • C21𝑒1𝑒
  • D81𝑒1𝑒
  • E41𝑒1𝑒

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