integral problem

Pace University MAT 236 CRN 22205 Prof. Kazlow Spring 2015 Take-home Quiz 3 March 25, 2015 Exam Instructions: This quiz is Due Monday May 5 in class. If you cannot be in class for any reason, please scan and submit your quiz to me by mail or submit via Blackboard before the beginning of class. You may use your book, notes and the internet to study the material covered by the exam. You may not use internet sources, CAS like Maple, Mathematica, nor Sage to supply you with the answers. Please submit your work together with the quiz. Only your quiz answers will be viewed. Your work will be checked only to verify that you did the work. Point values: 10 points each. Maximum Total is 120 points. 1. Find the local maximum and minimum values and note where they occur. Locate all saddle points: 2. Find the absolute/global maximum and minimum values of f on the set D. Note the points in the domain at which they occur: on 3. Use Lagrange multipliers to Yind the maximum and minimum values of the function subject to the constraint 4. Consider: deYined by and deYined by . Use the chain rule e to Yind the derivative of . 5. Evaluate , where 6. Express the integral in the reverse order (i.e., . 7. Find the volume of the solid bounded by the plane and the paraboloid by evaluating the integral where . f (x, y) = x 2 + xy + y 2 g(x, y) = 4x + 6y − x 2 − y 2 D = {(x, y)| 0 ≤ x ≤ 4,0 ≤ y ≤ 5} f (x, y) = x 2 y x 2 + 2y 2 = 6 f :!3 → ! f (x, y,z) = x 3 + 3xyz − y 2 z g :!2 → !3 g(s,t) = 2t + s,−t − s,t 2 + s 2 1− x 2 − y 2 ( )dA D ∫∫ D( f ! g)(1,2) y 2 dA D ∫∫ D = {(x, y)| −1 ≤ y ≤1,−2 − y ≤ x ≤ y} f (x, y)dy x2+1 5 ∫ ⎡ ⎣⎢ ⎤ ⎦⎥ dx 1 2 ∫ f (x, y)dx a b ∫ ⎡ ⎣⎢ ⎤ ⎦⎥ dy c d ∫ z = 0 z = 1− x 2 − y 2 D = {(x, y)| x 2 + y 2 ≤1} Page 1 of 2 Pace University MAT 236 CRN 22205 Prof. Kazlow Spring 2015 Take-home Quiz 3 March 25, 2015 9. Find the surface area of the part of the paraboloid that lies within the cylinder . 10. Find the limits of integration for calculating the volume of the solid Q enclosed by the graphs if is computed by repeated integration. 11. Find where R is the parallelogram with vertices . 12. Evaluate x = y 2 + z 2 y 2 + z 2 = 4 y = x 2 ,z = 0, y + z = 2 Volume = 1dV Q ∫∫∫ (4x + 8y) R ∫∫ dA (−1,−1),(1,3),(3,1),(5,5) cos(x + y + z)dz 0 x ∫ dx 0 y ∫ dy 0 π /2 ∫ Page 2 of 2