   Chapter 15.2, Problem 21E Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
ISBN: 9781285774770
Textbook Problem

Mass In Exercises 23 and 24, find the total mass of a spring with density ρ in the shape of the circular helix r ( t ) = 2 cos t i + 2 sin t j + t k , 0 ≤ t ≤ 4 π ⋅ ρ ( x , y , z ) = 1 2 ( x 2 + y 2 + z 2 )

To determine

To calculate: The total mass of a spring with density ρ if the spring has a density ρ(x,y,z)=12(x2+y2+z2) in the shape of the circular helix given by r(t)=2costi+2sintj+tk; 0t4π

Explanation

Given:

The spring has a density ρ(x,y,z)=12(x2+y2+z2) in the shape of the circular helix given by r(t)=2costi+2sintj+tk; 0t4π

Formula used:

For parameterization given by r(t)=xi+yj+zk, the length of curve ds is,

ds=r(t)dt

Calculation:

For the curve r(t)=2costi+2sintj+tk, x(t)=2cost, y(t)=2sint, and z(t)=t which implies that x(t)=2sint, y(t)=2cost, and z(t)=1. We can use x(t), y(t) and z(t) to find ds,

ds=r(t)dt=x(t)2+y(t)2+z(t)2dt=(2sint)2+(2cost)2+12dt=

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