x-t-cos-t-y-t-sin-t-0-t-pi-If-z-f-x-y-is-a-curtain-with-height-of-1-what-is-the-surface-area-of-the-curtain-

Question Number 2491 by Filup last updated on 21/Nov/15

x (t) = \cos t

y (t) = \sin t

0 ⩽ t ⩽ π

If : z = f (x, y) is a^{'} {c u r t a i n}^{'} with height

of 1, what is the surface area of the^{'} {c u r t a i n}^{'} ?

Commented by Yozzi last updated on 21/Nov/15

C u r t a i n w i t h h e i g h t 1 i m p l i e s t h a t

z = f (x, y) = 1 (0 ⩽ z ⩽ 1) a s x a n d y v a r i e s i n t h e x - y p l a n e .

F o r 0 ⩽ t ⩽ π i n t h e x - y p l a n e w e h a v e

a s e m i c i r c l e o f r a d i u s 1 s i n c e

x^{2} (t) + y^{2} (t) = {c o s}^{2} t + {s i n}^{2} t = 1 .

T h e a r c l e n g t h o f t h e c u r v e i s t h e n

s = r θ = 1 \times π = π .

T h e a r e a o n e i t h e r s i d e o f t h e c u r t a i n

c u r v e i s t h e s a m e s i n c e i t i s s i m p l y

t h e c u r v i n g o f t h e c r o s s - s e c t i o n o f a

v e r t i c a l r e c t a n g u l a r p l a n e w i t h

d i m e n s i o n s 1 b y π u m i t s . S o t h e t o t a l

s u r f a c e a r e a i s 1 \times 2 \times π = 2 π I t h i n k .

Answered by Yozzi last updated on 21/Nov/15

S . A = 2 π . T h i s a n s w e r a s s u m e s t h a t

t h e a r e a s o n t h e e d g e s o f t h e c u r t a i n

a r e a l l n e g l i g i b l e .

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