   Chapter 14.5, Problem 11E

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 ) = ln | sec x | R = { ( x , y ) : 0 ≤ x ≤ π 4 ,     0 ≤ y ≤ tan x }

To determine

To calculate: The area of the surface given by z=f(x,y)=ln|secx| which lies above the region R, which is represented by R={(x,y): 0xπ4,0ytanx}.

Explanation

Given: The surface is given by f(x,y)=ln|secx|, above the region R, which is represented by R={(x,y): 0xπ4,0ytanx}.

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)=ln|secx|.

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

fx(x,y)=ddx(ln|secx|)=1secxsecxtanx=tanx

Now, with respect to y

fy(x,y)=ddy(ln|secx|)=0

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

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