Exercise 1213Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Line integral on a piecewise path

#piecewise path#line integral#work

Statement

Compute $\int_C\vec F\cdot d\vec r$ for $\vec F=(x^2,y^2)$ along the broken line from $(0,0)$ to $(1,0)$ to $(1,1)$ .

Hint

Split into two segments; sum the contributions.

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Segment 1. From $(0,0)$ to $(1,0)$ : $\gamma_1(t)=(t,0)$ , $t\in[0,1]$ ; $\vec F\cdot\gamma_1'=t^2$ . Contribution $1/3$ .

Segment 2. From $(1,0)$ to $(1,1)$ : $\gamma_2(t)=(1,t)$ ; $\vec F\cdot\gamma_2'=t^2$ . Contribution $1/3$ .

Total. $2/3$ .

Conclusion. $\int_C\vec F\cdot d\vec r=2/3$ .