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August 21, 2022

We consider a region E of space delimited by a closed, piecewise smooth surface S.

We consider a region E of space delimited by a closed, piecewise smooth surface S.

a) Show that the volume of E can be calculated using the formula :

一、出。 V 1 3 (cz+ y+ zhi.ds. y 2 3 S

b) Use the formula shown in a) to calculate the volume of the region delimited by the area parameterized by

Ř(u, v) = (1+u) cos(u) 7 + (1 – u) sin(v) 7 +(1+u) k, -1 SuS1, -A SUST << -a < = -

and shown below :

2 1.5 2 1- 0.5- -2 - 1 -2 0 -1 0 - у 2 Х

c) The following question is independent of parts a) and b).
Show that the flux through a closed surface of a vector field of the form

F(x, y, z) = [ax + Aſy,z)] 7 + [by + B(x, z)]] + [cz +C(x, y)] k, = 2

where a, b, c are constants and A, B, C are functions, is proportional to the volume of the region inside the surface.

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