Exercise 593Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Line integral of a scalar field

#line integral#scalar field#parametrization

Statement

Compute $\displaystyle\int_C(x+y)\,ds$ where $C$ is the segment from $(0,0)$ to $(1,1)$ .

Hint

Parametrize $\gamma(t)=(t,t)$ , $t\in[0,1]$ , and $ds=\|\gamma'(t)\|\,dt$ .

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Parametrization. $\gamma(t)=(t,t)$ , $\gamma'(t)=(1,1)$ , $\|\gamma'(t)\|=\sqrt2$ .

Integral.
$\int_0^1(t+t)\sqrt2\,dt=2\sqrt2\int_0^1 t\,dt=\sqrt2.$

Conclusion. $\int_C(x+y)\,ds=\sqrt2$ .