{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Final_Fall-2008-2009_Kobeissi - American University of...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2008-2009 Final Exam Exercise 1 a. (5 points) If f ( u, v, w ) is a differentiable function and if u = x - y , v = y - z , and w = z - x , show that ∂f ∂x + ∂f ∂y + ∂f ∂z = 0 b. (10 points) Use the method of Lagrange multipliers to find the maximum and minimum values of f ( x, y ) = 3 x - y + 6 on the circle x 2 + y 2 = 4 Exercise 2 (10 points) Convert to polar coordinates, then evaluate the following integral Z 2 0 Z 0 - 1 - ( y - 1) 2 xy 2 dxdy Exercise 3 (12 points) Here is the region of integration of the integral = j x 6 z = y 2 y z (0,-1,1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1,-1,1) (1,-1,0) Z 1 0 Z 0 - 1 Z y 2 0 dz dy dx Rewrite the integral as an equivalent iterated integral in the other 5 orders, then evaluate one of them Exercise 4 Let V be the volume of the region D that is bounded by the paraboloid z = x 2 + y 2 , and the plane z = 2 y . a) (8 points) Express V as an iterated triple integral in cartesian coordinates in the order dz dx dy (do not evaluate the integral) . b) (10 points) Express V as an iterated triple integral in cylindrical coordinates, then evaluate the resulting integral. (you may use the result: R sin 4 x dx = - sin 3 x cos x 4 - 3 cos x sin x 8 + 3 x 8 )
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern