A wire lies along the piecewise linear curve extending from the point (4,3) to the point (18,5) to the point (18,11). If the density of the wire is given by p(x, y) = xy + 6x, use a line integral to find the mass of the wire. mass=

A thin hubcap is modeled by the part S of the sphere x² + y² + z² = 4 that lies above the plane z = 1, and is made of a material of constant density 1. What is the moment of inertia of S when it is rotated around the z-axis? (Use syntax like 2*pi/3 .)

A bird of mass m flying along a path given by r(t) = (2t, t^2, t) subject to the force of gravity pulling it towards the origin.  The force is F(x, y, z);  F(x, y, z) = (m/((x^2+y^2+z^2)^(3/2))(-x-y-z) How much work will the bird do fromt he time t=3 to t=8? Solve using the Fundamental Theorem of Line Integrals.

A wrench 1m long lies along the positive y-axis and grips a bolt at the origin (i.e. the wrench can be represented by vector <0,1,0>). A force is applied in the direction of the unit vector <0, 1/sqrt(2), 1/sqrt(2)> at the end of the wrench. Find the magnitude of the force in Newtons (N) needed to supply 5sqrt(2) N*m of torque to the bolt.  torque = | r x F |  if r = <0,1,0> and F = <0, 1/sqrt(2), 1/sqrt(2)> so the torque is equal to the magnitude of the cross product of these two vectors but I don't understand how to get incorporate the 5sqrt(2) N*m I know that the magnitude of the cross product must be equal to 5sqrt(2) but I am not sure how to work backwards to that answer, please help!.