   Chapter 16.2, Problem 1E

Chapter
Section
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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