   Chapter 16.2, Problem 2E

Chapter
Section
Textbook Problem

Evaluate the line integral, where C is the given curve.2. ∫C (x/y) ds, C: x = t3, y = t4, 1 ⩽ t ⩽ 2

To determine

To Evaluate: The line integral C(xy)ds for a curve C:x=t3,y=t4,1t2 .

Explanation

Given data:

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

C:x=t3,y=t4,1t2

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

Find the expression (xy) as follows.

Substitute t3 for x and t4 for y in the expression (xy) ,

xy=t3t4=1t

Evaluation of line integral C(xy)ds :

Substitute (xy) for f(x,y) , 1t for f(x(t),y(t)) , t3 for x , t4 for y , 1 for a , and 2 for b in equation (1),

C(xy)ds=121t(dt3dt)2+(dt4dt)2dt

Rewrite and compute the expression as follows.

C(xy)ds=121t[ddt(t3)]2+[ddt(t4)]2dt=121t(3t2)2+(4t3)2dt=121t9t4+16t6dt=121tt4(9+16t2)dt

Simplify the expression as follows

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