   Chapter 13.3, Problem 12E

Chapter
Section
Textbook Problem

# Find, correct to four decimal places, the length of the curve of intersection of the cylinder 4x2 + y2 = 4 and the plane x + y + z = 2.

To determine

To find: The length of C of the cylinder 4x2+y2=4 and the plane x+y+z=2 , with four decimal places.

Explanation

Given data:

Consider C is curve of intersection of cylinder 4x2+y2=4 and the plane x+y+z=2 .

The expression for cylinder is,

4x2+y2=4 (1)

Divide by 4 on both sides of equation (1).

14(4x2+y2)=44

x2+y24=1 (2)

The projection of the curve C onto the xy-plane is an ellipse 4x2+y2=4 or x2+y24=1 and z=0 .

Consider the parameter following parameters.

x=cost

y=2sint

Since C is also lies on the plane, the expression for the plane is,

x+y+z=2 (3)

Re-arrange equation (3).

z=2xy (4)

Substitute cost for x and 2sint for y in equation (4),

z=2cost2sint

The vector equation r(t) is,

r(t)=x,y,z

Substitute cost for x, 2sint for y, and 2cost2sint for z,

r(t)=cost,2sint,2cost2sint

Differentiate the equation with respect to t.

ddtr(t)=ddtcost,2sint,2cost2sintr(t)=ddt(cost),2ddt(sint),ddt(2cost2sint)r(t)=sint,2cost,sint2cost

Take magnitude on both sides

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