GATE Electrical Engineering (EE) 2019 Solved Paper

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Question : 35 of 65

Marks: +1, -0

If

f=2 x^{3}+3 y^{2}+4 z

, the value of line integral

∫↙{C}

grad

f

. dr evaluated over contour

C

formed by the segments

(-3,-3,2) →(2,-3,2) →(2,6,2) →(2,6,-1)

is___________.

Your Answer:

Solution:

Given

f=2 x^{3}+3 y^{2}+4 z

grad

f=i \;{∂ f}/{∂ x}+j \;{∂ f}/{∂ y}+k \;{∂ f}/{∂ z}

6 x^{2} i+6 y j+4 k

given that C is segment joining

(-3,-3,2) →(2,-3,2) →(2,6,2) →(2,6,-1)

∴ ∫↙{C}

grad.dr

=∫↙{C}(6 x^{2} i+6 y j+4 k) ⋅(d x i+d y j+d z k)

=∫↙{C}(6 {x}^{2} {dx}+6 {ydy}+4 {dz})

=∫↙{(-3,-3,2)}↖{(2,6,-1)}(6 {x}^{2} {dx}+6 {ydy}+4 {d} z)

equation of segment joining

(-3,-3,2)

and

(2,6,-1)

\;{x+3}/{2-(-3)}=\;{y+3}/{6-(-3)}=\;{z-2}/{-1-2}=t

⇒\table {x}=5 {t}-3 ⇒ {dx}=5 {dt} ;{y}=9 {t}-3 ⇒ {dy}=9 {dt} ; {z} =-3 {t}+2 ⇒ {d} {z}=-3 {dt}\}t=0 \to 1

=∫↙{t-0}↖{1} 6(5 t-3)^{2} 5 d t+6(9 t-3) 9 d t+4(-3) d t

=∫↙{t-0}↖{1} 30(25 t^{2}-30 t+9) d t+(486 t-162) d t-12 d t

[750 \;{t^{3}}/{3}-900 \;{t^{2}}/{2}+270 t+486 \;{t^{2}}/{2}-162 t-12 t]_{0}^{1}

[250-450+270+243-162-12]

=139

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