Question

Evaluate the following line integral along the curve C. ModifyingBelow Integral from nothing to nothing With Upper C (x plus y plus z )ds​; C is the part of the circle Bold r (t )equals left angle 5 cosine t comma 0 comma 5 sine t right angle​, for 0 less than or equals t less than or equals StartFraction pi Over 2 EndFraction .

Answer 1

If I'm reading this copy-pasted math right, you have $C$ parameterized by

$\vec r(t)=\langle5\cos t,0,5\sin t\rangle$

with $0\le t\le\frac pi2$. Then

$\vec r'(t)=\langle-5\sin t,0,5\cos t\rangle\implies\|\vec r'(t)\|=5$

so that

$\displaystyle\int_C(x+y+z)\,\mathrm dS=25\int_0^{\pi/2}(\cos t+\sin t)\,\mathrm dt=\boxed{50}$