Calculus Answers

Let r: [0,2)➝R^2 be the mapping defined by: r(t)=[x(t)=1/t*cos2t, y(t)=1/t*sin2t] and let t₀=

Notes: the toothpick bridge can hold upto 5lbs for 5 second and No part of the bridge may extend more than 10 cm below the surface on which the bridge is seated

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Imagine your wieght (5 lb) is 5 cm from one end, 25 cm from one end, 30 cm from one end, and 40 cm from one end. Find the forces the bridge applies to the supports under these conditions. Show your calculations using the cross product.

We know that the volume of a region E may often be computed as triple integral dxdydz over the region E. Show that the volume may also be computed as 1/3xsurface integral <x,y,z>.NdS where N is outward pointing unit normal to E

Let D be any surface and let F = <P(x),Q(y),R(z)> (P depends only on x, Q only on y,
and R only on z). Show that the surface integral F dot dr over the region D is equal to 0 (F · dr = 0.)

Evaluate the surface integral of <xz,yz,z> over the region D where D is given by z = a^2 − x^2 − y^2, x^2 + y^2 <= b2, oriented up.