- The line integral -V-dr- of the vector V-2xyz-x2z-x2yk- from the origin to the point 1- 1- 1

∫V⃗.d⃗r⃗=∫ ( 2xyzdx+x^2zdy+x^2ydz ) =∫_(0,0,0)^(1,1,1)d ( x^2yz )= ( x^2yz nbsp; )_(0,0,0)^(1,1,1) =1 0 = 1 Alternately: Equation of St. line x 0/1 0=y ...

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Engineering Mathematics

Line Integrals I

* The line in...

Question

* The line integral $\int \to V . d \to r$ of the vector $\to V = 2 x y z^i + x^{2} z^j + x^{2} y^k$ from the origin to the point (1, 1, 1)

is 1

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is zero

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is -1

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can not be determined without specifing the path

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Solution

The correct option is A is 1
$\int \to V . \to d r = \int (2 x y z d x + x^{2} z d y + x^{2} y d z)$
$= \int_{(0, 0, 0)}^{(1, 1, 1)} d (x^{2} y z) = {(x^{2} y z)}_{(0, 0, 0)}^{2}$
=1 - 0 = 1

Alternately:
Equation of St. line $\frac{x - 0}{1 - 0} = \frac{y - 0}{1 - 0} = \frac{z - 0}{1 - 0} = t (l e t)$
$\Rightarrow$ x = y = z = t so dx = dy = dz = dt and $0 \leq t \leq 1$
so, $\int_{c} \to V . d \to r = \int_{c} (2 x y z d x + x^{2} z d y + x^{2} y d z)$

$= \int_{0}^{1} 4 t^{3} d t = 1$

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* The line integral ∫→V.d→r of the vector →V=2xyz^i+x2z^j+x2y^k from the origin to the point (1, 1, 1)

* The line integral $\int \to V . d \to r$ of the vector $\to V = 2 x y z^i + x^{2} z^j + x^{2} y^k$ from the origin to the point (1, 1, 1)