Physics - W.E.P. Question with Solution | TestHub

The work done by a variable force $\vec{F}$ is given by $W=\int \vec{F}\cdot d\vec{r}$. Given $\vec{F}=(xy\hat{i}+y^2\hat{j})$ and $d\vec{r}=dx\hat{i}+dy\hat{j}$. Along the path $y=x$, we have $dy=dx$. Substituting $y=x$ into $\vec{F}$:$$\vec{F}=(x(x)\hat{i}+x^2\hat{j})=(x^2\hat{i}+x^2\hat{j})$$ $$W={\int }_{(0,0)}^{(a,a)}(x^{2}dx+x^{2}dy)$$ Since $dy=dx$ along the path: $$W={\int }_0^a(x^{2}dx+x^{2}dx)={\int }_0^{a}2x^{2}dx$$ $$W={\left[\frac{2x^3}{3}\right]}_0^a=\frac{2a^3}{3}-0=\frac{2a^3}{3}$$