   Chapter 13.1, Problem 44E

Chapter
Section
Textbook Problem

# Find a vector function that represents the curve of intersection of the two surfaces.44. The paraboloid z = 4x2 + y2 and the parabolic cylinder y = x2

To determine

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

Explanation

Given data:

The equations of paraboloid is z=4x2+y2 and parabolic cylinder is y=x2 .

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 y=x2 lies in xy-plane and z-plane is z=0 . Hence, consider x=t .

Substitute t for x in equation y=x2 ,

y=t2

Write the expression for surface of a paraboloid

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