   Chapter 16.2, Problem 3E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.3. ∫C xy4 ds, C is the right half of the circle x2 + y2 = 16

To determine

To Evaluate: The line integral C(xy4)ds for the right half of a circle x2+y2=16 .

Explanation

Given data:

The given curve C is a right half of the circle x2+y2=16 .

Formula used:

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

Cf(x,y)ds=abf(x(t),y(t))(dxdt)2+(dydt)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.

ddt(cost)=sintddt(sint)=cost[ddtf(t)][f(t)]ndt=[f(t)]n+1n+1

Write the equation of the circle as follows.

x2+y2=16

Consider the parametric equations such that the considered parameters must satisfy the equation of the circle x2+y2=16 .

Consider the parametric equations of the curve as follows.

x=4cost,y=4sint,π2tπ2

The considered parametric equations and the limits are satisfied if the parameters substitute in the equation of circle x2+y2=16 .

Find the expression (xy4) as follows.

Substitute 4cost for x and 4sint for y in the expression (xy4) .

xy4=(4cost)(4sint)4=45costsin4t

Evaluation of line integral C(xy4)ds :

Substitute (xy4) for f(x,y) , (45costsin4t) for f(x(t),y(t)) , 4cost for x , 4sint for y , (π2) for a , and π2 for b in equation (1),

C(xy4)ds=π2π2(45costsin4t)(ddt4cost)2+(ddt4sint)2dt

Rewrite and compute the expression as follows

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