MVC Chapter 6 Quiz NAME: Instructions: SHOW ALL WORK !! We don’t really need a calculator, do we? 2t 3 Let the curve C be given by: x(t ), y(t ), z (t ) t , t 2 , with 1 t 2 . Set up each 3 integral below as a single integral in terms of t. #1. a. xyz ds b. F d s given Fx, y, z = zi + yj + xk. C C #2. Find a parameterization for each of the following curves: a. The line segment from (1, 2, 4) to ( 2, 2, 0) b. The curve that starts at the point (0, 0, 0) and goes to the point (1, 0,1) along the circular paraboloid z x 2 y 2 . MVC #3. Let C be a curve in the plane starting at (0,0), moving to (1,1) along the curve y = x3, and then returning to the origin along the straight line y = x. a. Parameterize the path (in two pieces, most likely) to express the line integral 2 2 x y dx xy dy as an integral or integrals in the single variable t. DON’T integrate. C b. Apply Green’s Theorem to the integral in part a to obtain a double integral, making sure to provide appropriate limits of integration. DON’T integrate. MVC #3. (continued) Let C be a curve in the plane starting at (0,0), moving to (1,1) along the curve y = x3, and then returning to the origin along the straight line y = x. c. Given the vector field F( x, y) 2 x 2 y i xy j , write the integral(s) in the single variable t you would need to evaluate to find the outward flux of F across the curve C. DON’T integrate. d. Apply the divergence form of Green’s theorem to obtain a double integral that would calculate this flux. DON’T integrate. MVC #4 Use Green’s Theorem (with a thoughtful choice of F ( x, y ) M ( x, y )iˆ N ( x, y ) ˆj ) to set up a line integral that gives the area of the region inside the curve C : x(t ) t sin t , y (t ) 3cos t;0 t 2 . Write the line integral in terms of t. [Note that C is simple, smooth, closed curve]. #5. Set up a double integral that represents the area of the parametrized surface: X (2s 3t 1, t , s t ); 1 s 1,0 t 1 MVC x2 y 2 #6. Find a parametrization for the portion of the surface z that lies above the region in the x2 1 1 4 xy-plane bounded by the curves y x, y 3x, y , and y . . 3 x x MVC