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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find the area of the surface.

49. The surface with parametric equations xu2, y = uv, zimagesv2, 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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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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