2313-in-class-Oct12

2313-in-class-Oct12 - MAC 2313, Fall 2010 — In-class...

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Unformatted text preview: MAC 2313, Fall 2010 — In-class problems 10/12 These problems are solely for your edification, and will not be graded. You are encouraged to work on them in groups. 1 Use the method of Lagrange multipliers to find the maximum and minimum values of the function f (x, y ) = exy subject to the constraint x3 + y 3 = 16. 2 Calculate sin x dA, x R where R is the triangle in the xy -plane bounded by the x-axis, the line y = x, and the line x = 1. 3 Suppose that f (x, y ) is a continuous function. Sketch the region of integration for the following iterated integral and then reverse the order of integration. 1 π/4 f (x, y ) dy dx. 0 4 Some of the iterated integrals below correspond to real geometric problems (computation of volumes), but some do not. In fact, some are actually illegal. Indicate which are good and which are not. x2 y a. 0 0 5x 1 b. 0 x2 + y 2 + 1 dx dy , x2 + y 2 + 1 dx dy , 0 7 3y c. x2 + y 2 + 1 dx dy , y3 5 0 −y 2 d. −1 5 arctan x x2 + y 2 + 1 dx dy . 2y Suppose that f (x, y ) is a continuous function. We want to integrate f over the region R in the xy -plane lying between the circles x2 + y 2 = 4 and (x − 1)2 + y 2 = 1. Set up (but do not evaluate) the double integral f (x, y ) dA R as a sum of iterated integrals. ...
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