Exam_3_Solutions - Math 208 Midterm Exam 3 3/29/13 Key 1....

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Unformatted text preview: Math 208 Midterm Exam 3 3/29/13 Key 1. Set up an integral that computes the volume of a hemisphere of radius 4 through which has been drilled a circular hole of radius 1 along the axis both orthogonal to and passing through the center of its base. This object is radially symmetric around the axis through the center of the drilled hole, but not spherically symmetric, so cylindrical coordinates would be best. Take the z-axis to be the one through the center of the hole, and recall that the equation for the top half of a sphere in the z-direction is z = p 16- x 2- y 2 = √ 16- r 2 . Thus, the integral is Z 2 π Z 4 1 Z √ 16- r 2 d z d r d θ 2. Parameterize the circle of radius 3 in the ( x,y )-plane which is centered at the point (2,1), traversed clockwise. One clockwise parametrization of the unit circle is (cos( t ) ,- sin( t )). Here, we multiply by 3 to increase the radius and then shift the result. ~ r ( t ) = (3 cos( t ) + 2 ,- 3 sin( t ) + 1) , ≤ t ≤ 2 π 3. Consider the vector field ~ F = ( y 2 ,xy ). Is the parameterized curve ~ r ( t ) = (tan( t ) , sec( t )) a flow line of ~ F ?...
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