   Chapter 14.5, Problem 20E

Chapter
Section
Textbook Problem

Finding Surface AreaIn Exercises 17–20, find the area of the surface.The portion of the cone z = 2 x 2 + y 2 inside the cylinder x 2 + y 2 = 4

To determine

To calculate: The area of the surface given by z=f(x,y)=2x2+y2 which lies inside the cylinder x2+y2=4.

Explanation

Given: The surface is given by f(x,y)=2x2+y2, inside the cylinder x2+y2=4.

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

The equation of circle x2+y2=r2, where r is the radius

Calculation: The function given is f(x,y)=2x2+y2.

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

fx(x,y)=2ddx(x2+y2)=4x2x2+y2=2xx2+y2

Now, with respect to y,

It is found that:

fy(x,y)=2ddx(x2+y2)=4y2x2+y2=2yx2+y2

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

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