Math 215: Calculus 3, HW8 Due date: 11/29 or 12/1 1. Evaluate the integral 1 R ((x a)2 1/2 + y2 + z2 ) dV over the solid sphere 0 x2 + y 2 + z 2 1 for a > 1. Give a physical interpretation of your answer. 2. Let I = ex dx. Show that 2 2 e(x 2 I = +y 2 ) dx dy and use polar coordinates to evaluate I . This integral comes up in probability theory and statistics. 3. Find the center of mass of the quarter-circular ring x2 + y 2 = 1, x, y 0, assuming uniform density. 4. Evaluate the line integral C sin x dx + cos y dy when C consists of the top part of the circle x2 + y 2 = 1 from (1, 0) to (1, 0) followed by the line segment from (1, 0) to (2, 2). 5. Show that the line integral C (1 yex )dx + ex dy is independent of path. Evaluate for a path that begins at (1, 1) and ends at (3, 4). 6. Is the vector \u001celd F = sin yi + (1 + x cos y)j conservative? 7. Suppose F = f and f = sin(x2y) . Find a path C which is not closed such that and another path C such that C F.dr = 2. 8. Suppose F= r |r|3 where r = (x, y, z) is the position vector. Evaluate y = sin t, and z = 3t with 0 t 4 . 1 C C F.dr = 0 F.dr where C is the helix x = cos t