   Chapter 16.6, Problem 43E

Chapter
Section
Textbook Problem

Find the area of the surface.43. The surface z = (x3/2 + y3/2), 0 ⩽ x ⩽ 1, 0 ⩽ y ⩽ 1

To determine

To find: The area of the surface z=23(x32+y32),0x1,0y1 .

Explanation

Given data:

The equation of the surface is given as follows.

z=23(x32+y32),0x1,0y1

Formula used:

Write the expression to find the surface area of the plane.

A(S)=D1+(zx)2+(zy)2dA (1)

Write the equation of surface as follows.

z=23(x32+y32) (2)

Calculation of zx :

Take partial derivative for equation (2) with respect to x.

zx=x[23(x32+y32)]=23(32x321+0)=x12=x

Calculation of zy :

Take partial derivative for equation (2) with respect to y.

zy=y[23(x32+y32)]=23(0+32y321)=y12=y

Calculation of surface area of the plane:

Substitute x for zx and y for zy in equation (1),

A(S)=D1+(x)2+(y)2dA=D1+x+ydA=D(1+x+y)12dA

Apply the limits of x, y and rewrite the expression as follows

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