Application of Vector Calculus;Line integrals

Question

A force is given by $\ F=(cxy \ i + x^6y^2\ j)$, where $i$ and $j$ are unit vectors.The force acts on a particle which must move from (0,0) to the line x=1 along the curve $y=a(x^b)$ where $a>0,b>0$. Find a value of $a$ in terms of $c$ such that the work done is independent of $b$.

Answer 1

Use the definition of work $$\begin {align}W &= \int F\cdot dr \\ &= \int (cxy\ \hat{i} + x^6y^2\ \hat{j})\cdot(dx \ \hat{i} + dy\ \hat{j}) \\ &= \int cxy \ dx + x^6y^2\ dy \end{align}$$ You are given a curve $y = ax^b$ along which the particle moves. Use this to evaluate the integral.

(Substitute $y = ax^b$, $dy = ab\ x^{b-1} dx$)

$$\begin {align} W = \int_0^1 ac\ x^{b+1}\ dx + \int_0^1 ab \ x^{b+5} \ dx \end{align}$$