1
The Double Integral over a Rectangle- HW Problems
In problems 1-3, evaluate the iterated Integrals. First integrate with
respect to 𝑥 and then with respect to 𝑦. Then evaluate the integral by
reversing the order of integration. That is, integrate first with respect
to 𝑦 and then integrate with respect to 𝑥.
1 1
1. ∫0 ∫0 (6𝑥 2 𝑦 + 2𝑥 + 3𝑦 2 )𝑑𝑥𝑑𝑦
𝜋
𝜋
2. ∫0 ∫02 (sin(𝑥 ))(cos(𝑦))𝑑𝑥𝑑𝑦
1 2
3. ∫0 ∫1 (3𝑥 2 − 4𝑥𝑦)𝑑𝑥𝑑𝑦
4. Evaluate ∬𝑅 𝑥𝑒 𝑦 𝑑𝑦𝑑𝑥 where 𝑅 = [1,3] × [0, ln(2)].
5. Find the volume of the solid that lies over the rectangle
[1,2] × [0,2] and is bounded above by the following functions.
a. 𝑓(𝑥, 𝑦) = 30 − 3𝑥 2 − 3𝑦 2
b. 𝑓(𝑥, 𝑦) = 2 + 4𝑥 + 2𝑦
c. 𝑓(𝑥, 𝑦) = 3𝑥 2 + 3𝑦 2 .
The Double Integral over a Rectangle- HW Problems
In problems 1-3, evaluate the iterated Integrals. First integrate with
respect to 𝑥 and then with respect to 𝑦. Then evaluate the integral by
reversing the order of integration. That is, integrate first with respect
to 𝑦 and then integrate with respect to 𝑥.
1 1
1. ∫0 ∫0 (6𝑥 2 𝑦 + 2𝑥 + 3𝑦 2 )𝑑𝑥𝑑𝑦
𝜋
𝜋
2. ∫0 ∫02 (sin(𝑥 ))(cos(𝑦))𝑑𝑥𝑑𝑦
1 2
3. ∫0 ∫1 (3𝑥 2 − 4𝑥𝑦)𝑑𝑥𝑑𝑦
4. Evaluate ∬𝑅 𝑥𝑒 𝑦 𝑑𝑦𝑑𝑥 where 𝑅 = [1,3] × [0, ln(2)].
5. Find the volume of the solid that lies over the rectangle
[1,2] × [0,2] and is bounded above by the following functions.
a. 𝑓(𝑥, 𝑦) = 30 − 3𝑥 2 − 3𝑦 2
b. 𝑓(𝑥, 𝑦) = 2 + 4𝑥 + 2𝑦
c. 𝑓(𝑥, 𝑦) = 3𝑥 2 + 3𝑦 2 .
