GATE Civil Engineering (CE) 2020 Shift 1 Solved Paper

Question : 51 of 65

Marks: +1, -0

C

represents a line segment between

(0,0,0)

and

(1,1,1)

in Cartesian coordinate system, the value (expressed as integer) of the line integral

∫↙{c}[(y+z) d x+(x+z) d y+(x+y) d z]

is______________-.

Solution:

Equation of a line segment

\;{x}/{1}=\;{y}/{1}=\;{z}/{1}=k

x=k, y=k, z=k ⇒ d x=d k

0 ≤ x ≤ 1

∴ 0 ≤ k ≤ 1

∫↙{0}↖{1}(2 k) {d} k .+∫↙{0}↖{1 \/ 2} {k} {d} {k}+∫↙{0}↖{1} 2 {kdk}

=3[ {k}^{2}]_{0}^{1}

=3

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