Question

2. (8 points) A zone of a sphere is the portion of its surface which lies between two parallel planes which intersect the sphere. The altitude of the zone is the distance between the planes. Show that the surface area A of a zone depends only on its altitude h and the radius R of the sphere. [hint: you need to express A as a function of h and R.]

Answer 1

solution : fon Curve by alt) and y(t), the a parametesized акс. length is S- dy da 2 dt dt dt a from can the this desive we formula for the surface of anea Solid by a obtained Jotating this about the x-axis. thy turns out to be 2T Door de teatro > at of obtain Now the Zone de scired in the og questión by parametering X(t) Rcos (t) and 9 (t) = a sint 607 Osts 4 ६०. de - R sinlt), at dy RCOS s(1) at

A. CT sint? Jer since + (Rcast)? dt 21 a sint dt 211 R2 siniti dt o 2 Jinct) dt 2TR u e CITR -cost st] Coso الا cos o Zur? Rcos (0) RCOS (0) Nos h = R I Coso - cosa A. (25R) R { coso cosa] RIR h. AD B thank you, sle rate.