Given names student number signature for markers only

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GIVEN NAMES: STUDENT NUMBER: SIGNATURE: FOR MARKERS ONLY 1. /10 2. /10 3. /10 4. /10 5. /10 6. /15 7. /10 8. /10 9. /15 TOTAL /100 –2–

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10 1. Give an integral which is equivalent to integraltext 4 0 integraltext y y ( x - y 2 ) dx dy with the order of integration reversed. Note: EVALUATION OF THE INTEGRAL IS NOT REQUIRED. 10 2. Compute integraltext B xy dA where B is the region cut off from the first quadrant by the parabola y = 1 - x 2 . –3–
10 3. Compute integraltext 1 0 integraltext 1 x x cos ( π 2 y 3 ) dy dx . 10 4. Give an integral in the polar variables r and θ which is equivalent to integraldisplay 4 0 integraldisplay 4 x - x 2 0 radicalbig x 2 + y 2 dy dx. Note: EVALUATION OF THE INTEGRAL IS NOT REQUIRED. –4–

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10 5. Find the volume of the figure in the first octant bounded by z = 4 - y 2 and y = 2 x . –5–
15 6. Find the volume of the figure bounded below by the paraboloid z = x 2 + y 2 and bounded above by the plane z = 2 y . –6–

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10 7. Compute integraltext γ F · ds where F ( x, y, z ) = ( z 2 , e x , e y ) and γ ( t ) = (1 , t, e t ), 0 t 1. 10 8. Compute integraltext γ dx + x dy + y dz where γ ( t ) = (cos t, sin t, t ), 0 t 2 π . –7–
15 9. A bicycle whose wheels have radius 1 m. is moving in a straight line at 1 m./sec. Find the formula which gives the position at time t of the point which is initially at the top of the wheel. What is the total distance travelled by this point during the first revolution? (Assume that the coordinate system is chosen so that the center of the wheel moves from left to right along the x -axis starting from the origin.) –8–

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