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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.

1.C y ds, C: x = t2, y = 2t, 0 ⩽ t ⩽ 3

To determine

To Evaluate: The line integral Cyds for a curve C:x=t2,y=2t,0t3 .

Explanation

Given data:

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

C:x=t2,y=2t,0t3

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.

ddttn=ntn1[ddtf(t)][f(t)]ndt=[f(t)]n+1n+1

Evaluation of line integral Cyds :

Substitute y for f(x,y) , 2t for f(x(t),y(t)) , t2 for x , 2t for y , 0 for a , and 3 for b in equation (1),

Cyds=032t(dt2dt)2+(d2tdt)2dt

Rewrite and compute the expression as follows.

Cyds=032t[ddt(t2)]2+(2dtdt)2dt=032t(2t)2+(2)2dt=032t4t2+4dt=032t4(t2+1)dt

Simplify 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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