   Chapter 16.6, Problem 49E

Chapter
Section
Textbook Problem

Find the area of the surface.49. The surface with parametric equations x = u2, y = uv, z = v2, 0 ⩽ u ⩽ 1, 0 ⩽ v ⩽ 2

To determine

To find: The area of surface with the parametric equations x=u2,y=uv,z=12v2,0u1,0v2 .

Explanation

Given data:

The parametric equations of the surface are given as follows.

x=u2,y=uv,z=12v2,0u1,0v2

Formula used:

Write the expression to find the surface area of the plane with the vector equation r(u,v) .

A(S)=D|ru×rv|dA (1)

Here,

ru is the derivative of vector equation r(u,v) with respect to the parameter u and

rv is the derivative of vector equation r(u,v) with respect to the parameter v .

Write the expression to find ru .

ru=u[r(u,v)] (2)

Write the expression to find rv .

rv=v[r(u,v)] (3)

Write the parametric equations of the surface as follows.

x=u2,y=uv,z=12v2

Write the vector equation of the surface from the parametric equations.

r(u,v)=u2i+uvj+12v2k

Calculation of ru :

Substitute u2i+uvj+12v2k for r(u,v) in equation (2),

ru=u(u2i+uvj+12v2k)=u(u2)i+u(uv)j+u(12v2)k=2ui+vj+(0)k

Calculation of rv :

Substitute u2i+uvj+12v2k for r(u,v) in equation (3),

rv=v(u2i+uvj+12v2k)=v(u2)i+v(uv)j+v(12v2)k=(0)i+uj+12(2v)k=(0)i+uj+vk

Calculation of ru×rv :

Substitute 2ui+vj+(0)k for ru and (0)i+uj+vk for rv in the expression ru×rv ,

ru×rv=(2ui+vj+(0)k)×((0)i+uj+vk)

Rewrite and compute the expression as follows

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