   Chapter 16.2, Problem 9E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.9. ∫C x2 y ds, C: x = cos t, y = sin t, z = t, 0 ⩽ t ⩽π/2

To determine

To Evaluate: The line integral C(x2y)ds for a curve.

Explanation

Given data:

The parametric equations of curve and its limits are given as follows.

C:x=cost,y=sint,z=t,0tπ2

Formula used:

Write the expression to evaluate the line integral for a function f(x,y,z) along the curve C .

Cf(x,y,z)ds=abf(x(t),y(t),z(t))(dxdt)2+(dydt)2+(dzdt)2dt (1)

Here,

a is the lower limit of the curve C and

b is the upper limit of the curve C .

Write the required differential and integration formulae to evaluate the given integral.

ddttn=ntn1ddtcost=sintddtsint=cost[ddtf(t)][f(t)]ndt=[f(t)]n+1n+1

Calculation of expression (x2y) :

Substitute cost for x and sint for y in the expression (x2y) ,

x2y=(cost)2(sint)=cos2tsint

Evaluation of line integral C(x2y)ds :

Substitute (x2y) for f(x,y,z) , (cos2tsint) for f(x(t),y(t),z(t)) , cost for x , sint for y , t for z , 0 for a , and π2 for b in equation (1),

C(x2y)ds=0π2(cos2tsint)(ddtcost)2+(ddtsint)2

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