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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 .

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LammettHash

If I'm reading this copy-pasted math right, you have [tex]C[/tex] parameterized by

[tex]\vec r(t)=\langle5\cos t,0,5\sin t\rangle[/tex]

with [tex]0\le t\le\frac pi2[/tex]. Then

[tex]\vec r'(t)=\langle-5\sin t,0,5\cos t\rangle\implies\|\vec r'(t)\|=5[/tex]

so that

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