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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find a vector function that represents the curve of intersection of the two surfaces.

42. The cylinder x2: + y2 = 4 and the surface z = xy

To determine

To find: the vector function of the curve that intersects two surfaces.

Explanation

Given data:

The equations of cylinder as x2+y2=4 and surface as z=xy .

Definition:

Consider parametric equations as x=f(t) , y=g(t) , and z=h(t) .

Write the expression for vector function in terms of parametric equations.

r(t)=f(t)i+g(t)j+h(t)k (1)

Here,

f(t) , g(t) , and h(t) are components functions of r(t) , and

x, y, and z are parametric equations of space curve C.

The equation of cylinder is x2+y2=4 indicates that the z-plane is z=0 . Consider the value of x as 2cost and y as 2sint for cylinder surface.

Substitute 2cost for x and 2sint for y in cylinder equation x2+y2=4 ,

(2cost)2+(2sint)2=44cos2t+4sin2t=44(cos2t+sin2t)=44(1)=4 {cos2x+sin2x=1}

4=4

The right hand side (RHS) is equal to the left hand side (LHS)

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Chapter 13 Solutions

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