Exercise 1214Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Work done by a force field

#work#line integral#vector field

Statement

Compute the work of $\vec F(x,y)=(y,x)$ along the segment from $(0,0)$ to $(1,2)$ .

Hint

Work $=\int_C\vec F\cdot d\vec r$ . Parametrize $\gamma(t)=(t,2t)$ .

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Parametrization. $\gamma(t)=(t,2t)$ , $\gamma'(t)=(1,2)$ , $t\in[0,1]$ .

Integrand. $\vec F(\gamma(t))=(2t,t)$ , $\vec F\cdot\gamma'=2t+2t=4t$ .

Integral. $\int_0^1 4t\,dt=2$ .

Conclusion. Work $=2$ .