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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Lateral Surface Area In Exercises 65-72, find the area of the lateral surface (see figure) over the curve C in the x y-plane and under the surface z = f ( x , y ) , where

Lateral surface area = C f ( x , y ) d s .

images

f ( x , y ) = x y , C : y = 1 x 2 from (1, 0) to (0, 1)

To determine

To Calculate: The area of the lateral surface over the curve C in the xy-plane and under the given surface z=f(x,y)=h where lateral surface area is Cf(x,y)ds. The function is f(x,y)=xy.

Refer to figure in question.

Explanation

Given:

The given surface is f(x,y)=xy and C:y=1x2 from (1,0) to (0,1).

Formula Used:

The integration formula is u2+a2du=12(uu2+a2+a2ln|u+u2+a2|)+C.

And,

u2u2+a2du=18[u(2u2+a2)u2+a2+a4ln|u+u2+a2|]+C

Calculation:

Let x=cost then,

y=1cos2t=sin2t

Usingthis data, the position vector as:

r(t)=xi+yj=costi+sin2tj,0tπ2

On differentiating r(t),

r(t)=sinti+2sintcostjr(t)=sin2t+4sin2tcos2t=sint1+4cos2t

So,

ds=r(t)dt=sint1+4cos2tdt

Thus, lateral surface area,

Cf(x,y)ds=Cxyds=0π2costsin2tsint

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