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Unformatted text preview: ESP Kouba Worksheet 6 1.) Assume that w = f(x,y,z) and f(u —— t, t, u) = 0. Show that fy + fz = 0. 2.) Evaluate the following double integrals. ’E’ — c050 a.) Ff 3r2sec0drd0 0 0 1 -—x 2 2 b.)f f ex+y dydx 0 0 1 x 0).}; fLsz'ty (1de fl 3.) Assume that region R is described in polar coordinates by a 5 0 3 fl and 0 g r 5 f( 0). Show that the area of region R is Area = § 1' fl [f(0)]2 d0. 4.) Consider the cylinder above the circle (x —- $2 + y2 = i in the xy — plane and below the plane 2 = x + 1. Compute its volume. 5.) Determine an equation for the plane tangent to the surface 2 = x2+ y4 at the point (1, —1, 2). I 6.) A thin lamina lies in the triangular region with vertices (0,0), (0,2), and (3,2). Density at point (x,y) is f(x,y) = x2 + y. Set up but do not evaluate the integrals which represent a.) its centroid. b.) its center of mass. c.) the moment about i.) the line x=1. ii.) the line y=2. d.) the moment of inertia about i.) the origin ii.) the line it = 4. 7.) Draw the solids described below. a.) —2$x§0,——4_x gyg 4—x, 032312 + y? b.) £50$%,0$r$2,r2$z$4 8.) Use rectangular coordinates to describe the tetrahedron with corners (2,0,2), (1,0,2), (1,1,2), and (1,1,1). a.) First project it onto the xy —- plane. b.) First project it onto the xz — plane. 9.) Let R bethe solid prism with vertices (0,0,0), (0,0,1), (0,2,0), (0,2,1), (3,0,0), and (3,0,1). Evaluate f 1 dv. What does your answer represent? R 10.) Compute f z dv, where R is the region above the rectangle whose vertices are R (0,0,0), (2,0,0), (2,3,0), and (0,3,0) and below the plane z = x + 2y. ...
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