1 Assume S is a solid in three space lying entirely between

Last updated: 10/24/2023

1 Assume S is a solid in three space lying entirely between the planes x a and x b each of whose x cross sections S is a circle of radius r x where r is a continuous function Fix positive scaling factors a and 6 and let T be the solid whose points are of the form ax By Bz where P x y z is a point of S In other words T is obtained by stretching S by an x scaling factor of a in the x direction and by a yz scaling factor of in the y and z directions a Using the volume via cross sections definition derive a formula for vol T in terms of vol S a and B Hint describe the x cross sections T in terms of the x cross sections of S What kind of shapes are they How do their dimensions relate to the dimensions of the cross sections of S Suggestion it may help to do the sketch in part b to help you visualize what is going on b Consider the particular example where S is the solid sphere of radius 1 centered at the origin and T is the solid obtained by stretching S as above with the x scaling factor a 4 and the yz scaling factor 2 i Provide a sketch of T that includes x y and z axes with scales Hint start with 6 easy to plot points on the sphere S and determine what happens to them when you stretch by c and d Computo