   Chapter 13.3, Problem 11E

Chapter
Section
Textbook Problem

# Let C be the curve of intersection of the parabolic cylinder x2 = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36).

To determine

To find: The exact length of C from the origin to the point (6,18,36) .

Explanation

Given data:

Consider C is curve of intersection of parabolic cylinder x2=2y and surface 3z=xy .

Consider the expression for parabolic cylinder.

x2=2y (1)

Re-arrange equation (1),

y=12x2 (2)

The projection of the curve C onto the xy-plane is the curve x2=2y or y=12x2 and z=0 .

Choose the parameter,

x=t

Substitute t for x in equation (2),

y=12t2

Consider the expression for the surface.

3z=xy (3)

Re-arrange equation (3).

z=13xy (4)

Substitute t for x and 12t2 for y in equation (4),

z=13(t)(12t2)=16t3

The vector equation r(t) is,

r(t)=x,y,z (5)

Substitute t for x, 12t2 for y, and 16t3 for z in equation (5),

r(t)=t,12t2,16t3 (6)

The origin point (0,0,0) obtained for t=0 and the point (6,18,36) is obtained for t=6 .

Differentiate equation (6) with respect to t.

r(t)=1,22t,36t2=1,t,12t2

Write the expression for the extract length of C from the origin of the point (6,18,36)

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