x3a_sols - Fall 2010 Math 251 Exam 3A: Solutions Tue,...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Fall 2010 Math 251 Exam 3A: Solutions Tue, 09/Nov c 2010 Art Belmonte Refer to diagrams drawn on your exam. 1. Find the volume of the solid in the first octant bounded by the circular cylinders x 2 + z 2 = 9 and y 2 + z 2 = 9, and the coordinate planes. Rectangular coordinates are recommended. Use the quarter-circular region in the yz-plane as your shadow region. (Alternatively, use a similar region in the xz-plane or a square in the xy-plane.) The volume is V = ZZZ E 1 dV = Z 3 Z 9- y 2 Z 9- z 2 1 dxdzdy = 18 cm 3 . 2. Evaluate the integral Z ln10 Z 10 e x 1 ln y dydx by reversing the order of integration. Express the curve y = e x as x = ln y . Then Z 10 1 Z ln y 1 ln y dxdy = 9. 3. Determine the mass and center of mass of the finite region in the xy-plane bounded by the curves y = 1 1 + x 2 and y = 1 2 x 2 if its density is = y . When the curves intersect, their y-coordinates are equal. Thus 1 1 + x 2 = 1 2 x 2 , whence x = 1....
View Full Document

Ask a homework question - tutors are online