Line integral problems in Apostol calculus

[Richardbryant]

Homework Statement

A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0

Homework Equations

Find a value of a(in terms of c) such that the work done by this force is independent of b

The Attempt at a Solution

I pluck the information x=1 into y=ax^b which gives y=a,so i believe the curve move from (0,0) to (1,a)
then i parametrize the curve as r(t)=ti+at^bj which give r'(t)=i+abt^(b-1)j
With plucking x=t, y=at^b into f(x,y) with the upper and lower limit in the integral, the solution i got is
ac/(b+2)+a^(3)b/(b+18)

However, the solution from the book is a=(3c/2)^(1/2)

May i know which of my steps are correct and wrong, and teach me the right way of doing this question?

 

[pasmith]

Homework Statement

A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0

Homework Equations

Find a value of a(in terms of c) such that the work done by this force is independent of b

The Attempt at a Solution

I pluck the information x=1 into y=ax^b which gives y=a,so i believe the curve move from (0,0) to (1,a)
then i parametrize the curve as r(t)=ti+at^bj which give r'(t)=i+abt^(b-1)j
With plucking x=t, y=at^b into f(x,y) with the upper and lower limit in the integral, the solution i got is
ac/(b+2)+a^(3)b/(b+18)

I get the line integral as $$W = \frac{ac}{b+2} + \frac{a^3b}{6 + 3b}.$$ I can only assume that you did not multiply the $\mathbf{j}$ components correctly or did not correctly integrate the result; as you haven't actually shown that working I can't help you.

However, the solution from the book is a=(3c/2)^(1/2)

May i know which of my steps are correct and wrong, and teach me the right way of doing this question?

You have yet to finish the question: how do arrange that $W$ (which is a quadratic in $b$ whose coefficients are functions of $a$ and $c$ divided by a quadratic in $b$ whose coefficients are known constants) is independent of $b$?