i would greatly appreciate any help... A cylinder of unspecified radius r and he

Question

i would greatly appreciate any help...

A cylinder of unspecified radius r and height h runs along the z-axis. Parametrize the lateral surface of the cylinder using cylindrical coordinates. Find ||T0 Times Tz\ = Derive the surface area formula 2 pi rh by finding Integrate f(x, y, z) = xyz over the surface in Problem 1. A surface S is parametrized by Set up and simplify the integral Do not integrate. You will, of course, need to find ||Tu Times Tv||. S is the upper hemisphere given by x2 + y2 + z2 = 1 where z 0. Parametrize the surface using spherical coordinates. Find ||T Times T theta|| Set up and simplify the integral Do not integrate. Integrate dS where F(x, y, z) = (y, x, 4). (use F mid dot T Times T theta = F mid dot (x, y, z)rho sin )

Explanation / Answer

1) lateral surface(theta,z) = r*theta*z where -pi<theta<pi and 0<z<h

surface area = integral(ds) = integral(r*dtheta*dz) = R*2pi*h

2) integral(xyz) = integral(r*xyz*dtheta*dz) = integral(r^3sin(theta)cos(theta)z*dtheta*dz) = 0 since sin is an odd function so its integral in one period is zero.

4) a) lateral surface(theta,phi) = r^2*sin(theta)*d(theta)*d(phi) where -pi<phi<pi and 0<theta<pi

c) integral(x^2 + y^2 + z^4) = integral({ [r*cos(phi)sin(theta)]^2 + [r*sin(phi)sin(theta)]^2 + [r*cos(theta)]^2} r*sin(theta)*d(theta)*d(phi))

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