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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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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Chapter 16 Solutions

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Sect-16.1 P-11ESect-16.1 P-12ESect-16.1 P-13ESect-16.1 P-14ESect-16.1 P-15ESect-16.1 P-16ESect-16.1 P-17ESect-16.1 P-18ESect-16.1 P-21ESect-16.1 P-22ESect-16.1 P-23ESect-16.1 P-24ESect-16.1 P-25ESect-16.1 P-26ESect-16.1 P-29ESect-16.1 P-30ESect-16.1 P-31ESect-16.1 P-32ESect-16.1 P-33ESect-16.1 P-34ESect-16.1 P-35ESect-16.1 P-36ESect-16.2 P-1ESect-16.2 P-2ESect-16.2 P-3ESect-16.2 P-4ESect-16.2 P-5ESect-16.2 P-6ESect-16.2 P-7ESect-16.2 P-8ESect-16.2 P-9ESect-16.2 P-10ESect-16.2 P-11ESect-16.2 P-12ESect-16.2 P-13ESect-16.2 P-14ESect-16.2 P-15ESect-16.2 P-16ESect-16.2 P-17ESect-16.2 P-18ESect-16.2 P-19ESect-16.2 P-20ESect-16.2 P-21ESect-16.2 P-22ESect-16.2 P-23ESect-16.2 P-24ESect-16.2 P-25ESect-16.2 P-26ESect-16.2 P-31ESect-16.2 P-32ESect-16.2 P-33ESect-16.2 P-34ESect-16.2 P-35ESect-16.2 P-36ESect-16.2 P-37ESect-16.2 P-38ESect-16.2 P-39ESect-16.2 P-40ESect-16.2 P-41ESect-16.2 P-42ESect-16.2 P-43ESect-16.2 P-44ESect-16.2 P-45ESect-16.2 P-46ESect-16.2 P-47ESect-16.2 P-48ESect-16.2 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