   Chapter 14.5, Problem 9E

Chapter
Section
Textbook Problem

Finding Surface AreaIn Exercises 3–16, find the area of the surface given by z = f ( x , y ) that lies above the region R . f ( x , y ) = 3 + 2 x 3 / 2 R: rectangle with vertices ( 0 , 0 ) ,     ( 0 , 4 ) ,     ( 1 , 4 ) ,     ( 1 , 0 )

To determine

To calculate: The area of the surface given by z=f(x,y)=3+2x32 which lies above the region R, which is rectangular with vertices (0,0),(0,4),(1,4) and (1,0).

Explanation

Given: The surface is given by f(x,y)=3+2x32, above the region R, which is rectangular with vertices (0,0),(0,4),(1,4) and (1,0).

Formula used: The surface area can be calculated of the region R by,

S=R1+[fx(x,y)]2+[fy(x,y)]2dA

Differentiation formula ddx(xn)=nxn1,ddx(constant)=0

Calculation: The function given is f(x,y)=3+2x32.

Now partially differentiating it with respect to x, use ddx(xn)=nxn1,ddx(constant)=0

fx(x,y)=ddx(3+2x32)=2(32)x322=3x12

Now, with respect to y.

fy(x,y)=ddy(3+2x32)=0

Substitute in the formula S=R1+[fx(x,y)]2+[fy(x,y)]2dA

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