CALCULUS III
FINAL EXAM
Sample 3
Student Name: ___________________________________________
Student ID Number: ______________________________________
Course Title: Calculus III
Duration of Exam: 2 Hours
Number of Exam Pages: 7 pages (including this cover sheet)
Exam Type: Closed book
Total Mark: /30
,Question 1 (4 marks)
Sketch the region of integration and express the following sum as a single iterated integral by
interchanging the order of integration:
1 𝑦 2 2−𝑦
∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 + ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
0 0 1 0
, Question 2 (5 marks)
Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar
integral.
𝟏 √𝟏−𝒚𝟐
∫ ∫ 𝐥𝐧(𝒙𝟐 + 𝒚𝟐 + 𝟏)𝒅𝒙 𝒅𝒚
−𝟏 −√𝟏−𝒚𝟐
FINAL EXAM
Sample 3
Student Name: ___________________________________________
Student ID Number: ______________________________________
Course Title: Calculus III
Duration of Exam: 2 Hours
Number of Exam Pages: 7 pages (including this cover sheet)
Exam Type: Closed book
Total Mark: /30
,Question 1 (4 marks)
Sketch the region of integration and express the following sum as a single iterated integral by
interchanging the order of integration:
1 𝑦 2 2−𝑦
∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦 + ∫ ∫ 𝑓(𝑥, 𝑦)𝑑𝑥 𝑑𝑦
0 0 1 0
, Question 2 (5 marks)
Change the Cartesian integral into an equivalent polar integral. Then evaluate the polar
integral.
𝟏 √𝟏−𝒚𝟐
∫ ∫ 𝐥𝐧(𝒙𝟐 + 𝒚𝟐 + 𝟏)𝒅𝒙 𝒅𝒚
−𝟏 −√𝟏−𝒚𝟐
