3 Use cylindrical coordinates in the following problems (a) Evaluate RRR E p x2 y2 dV , where E is the solid bounded by the paraboloid z = 9 − x 2− y and the xyplane Solution In cylindrical coordinates the region E is described by11 Consider the paraboloid z = x2 y2 (a) Compute equations for the traces in the z = 0, z = 1, z = 2, and z = 3 planes Plane Trace z = 0 Point (0;0) z = 1 Circle x 2 y = 1 z = 2 Circle x 2 y = 2 z = 3 Circle x2 y2 = 3 (b) Sketch all the traces that you found in part (a) on the same coordinate axesFigure 6 Graph of the elliptic paraboloid z = x 2 4 y 9 De nition The quadric surface de ned by z c = x 2 a 2 y b is called a hyperbolic paraboloid Its traces in vertical planes x = k or y = k are parabolas and its traces in horizontal planes z = k are hyperbolas Figure 7 Graph of the hyperbolic paraboloid z = x 2 4 y 9
Multiple Integrals H 2 Y Are Continuous Functions On C D And Let F X Y Be A Function Defined On R Then Pdf Free Download
