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MAT 272 Test 3 Review.
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MAT 272 Test 3 Review13.1 Double Integrals over Rectangular Regions
1. Compute a Riemann sum approximation of
∬ 𝑓(𝑥, 𝑦)𝑑𝐴
𝐷
where 𝐷 = [−1,1]2 (the square of all points (x,y) with −1 ≤ 𝑥 ≤ 1, −1 ≤ 𝑦 ≤
1), based on the following information:
1 1 1 1 1 1 1 1
𝑓 (− 2 , − 2) = 1, 𝑓 (2 , 2) = 2, 𝑓 (2 , − 2) = 3, 𝑓 (− 2 , 2) = 4.
a. 10
b. 2.5
c. 40
d. 20
e. None of the above.
2. True or false? The double integral of a continuous function 𝑓 over the
rectangular region 𝐷 = [𝑎, 𝑏] × [𝑐, 𝑑] can be computed in two ways as
𝑏 𝑑 𝑑 𝑏
∬ 𝑓(𝑥, 𝑦)𝑑𝐴 = ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥 = ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 .
𝐷 𝑎 𝑐 𝑐 𝑎
3. True or false? The double integral of 𝑓(𝑥, 𝑦) = 𝑔(𝑥)ℎ(𝑦), where 𝑔 is
continuous on [𝑎, 𝑏] and ℎ is continuous on [𝑐, 𝑑] over the rectangular region
𝐷 = [𝑎, 𝑏] × [𝑐, 𝑑] can be computed as
𝑏 𝑑
∬ 𝑓(𝑥, 𝑦)𝑑𝐴 = ∫ 𝑔(𝑥)𝑑𝑥 ∫ ℎ(𝑦)𝑑𝑦.
𝐷 𝑎 𝑐
1 2
4. Evaluate ∫0 ∫0 𝑥 2 𝑦 3 𝑑𝑦 𝑑𝑥 efficiently.
, MAT 272 TEST 3 and Final Exam Review
13.2 Double Integrals over General Regions
1. Calculate ∬𝑅 2𝑥𝑦𝑑𝐴 where R is the region between the curves 𝑦 = √𝑥 and
1
𝑦 = 3 𝑥.
2. Reverse the order of integration on the following double integral.
4 √𝑥
∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑦 𝑑𝑥
1 1
𝑦 4
a. ∫1√ ∫1 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
2 𝑦2
b. ∫1 ∫0 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
𝑦 2
c. ∫1√ ∫1 𝑓(𝑦, 𝑥)𝑑𝑥 𝑑𝑦
2 4
d. ∫1 ∫𝑦 2 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
e. None of the above.
3. Evaluate the integral. Show all your steps.
1 1
3
∫ ∫ 𝑒 𝑥 𝑑𝑥 𝑑𝑦
0 √𝑦
13.3 Double Integrals in Polar Coordinates
1. Evaluate the iterated integral by converting to polar coordinates:
2
√4−𝑥 2
2 +𝑦 2
∫∫ 𝑒𝑥 𝑑𝑦𝑑𝑥
0
−2
𝜋
a. (𝑒 − 1) 4
2
b. 𝜋𝑒 4
𝜋
c. 𝑒4
4
d. 2𝜋(𝑒 4 − 1)
e. None of the above.
© 2016 School of Mathematical Sciences and Statistics, Arizona State University 2
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