2313-practice-mid4 - MAC 2313 Fall 2010 Midterm 4 Review...

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MAC 2313, Fall 2010 — Midterm 4 Review Problems 1 We will discuss these problems in class on Monday 11/15. Solutions will be posted on the course webpage over the weekend. 1 A bounded region in the first octant of 3-dimensional Euclidean space has the surface x + y + z 2 = 1 as part of its boundary. The remainder of its boundary is given by portions of the places x = 0, y = 0, and z = 0. Compute the triple integral of z over this region in space. (You should probably begin by sketching this region.) 2 Let R denote the portion of the unit ball { ( x, y, z ) : x 2 + y 2 + z 2 1 } that lies inside the solid cone { ( x, y, z ) : z radicalbig x 2 + y 2 } . Compute the volume of R with triple integrals in both cylindrical and spherical coordinates. 3 Let R denote the portion of the unit ball { ( x, y, z ) : x 2 + y 2 + z 2 1 } that lies inside the prism { ( x, y, z ) : x 0, y 0, and x + y 1 } . Set up a triple integral to compute the volume of R . 4 Find the work done by the field F = y i x j + k as a particle is moved from (1 , 0 , 0) to (1 , 0 , 1) along the path r ( t ) = ( cos t, sin
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