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Chapter 14: Q 27. (page 1106)

Evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y, z) = e^{x^{2} + y + z^{2}}$ with $C$ the circular helix of radius $1$ , centered about the z-axis, and parametrized by $r (t) = (\cos t, \sin t, t)$ from height $0$ to $蟺$ .

Short Answer

Expert verified

The required integral is ${鈭�}_{C} (e^{x^{2} + y + z^{2}}) d s = \sqrt{2} e (e^{蟺} - 1)$

Step by step solution

01

Given Information

It is given that $f (x, y, z) = e^{x^{2} + y + z^{2}}$

Curve is parametrized by

$r (t) = 鉄� \cos t, \sin t, t 鉄�$

02

Solving the parametrized curve

The parametrized curve is $x (t) = \cos t, y (t) = \sin t, z (t) = t, 0 鈮� t 鈮� 蟺$

$鈬� x^{'} (t) = - \sin t, y^{'} (t) = \cos t, z^{'} (t) = 1$

Also

$d s = ||r^{'} (t)|| d t$

$= \sqrt{{[x^{'} (t)]}^{2} + {[y^{'} (t)]}^{2} + {[z^{'} (t)]}^{2}} d t$

$= \sqrt{(- \sin t)^{2} + (\cos t)^{2} + (1)^{2}} d t$

$= \sqrt{1 + 1} d t$

$= \sqrt{2} d t$

$鈬� f (x (t), y (t), z (t)) = e^{\cos^{2} t + \sin t + t^{2}}$

03

Solving the line integral

The form of line integral is

${鈭�}_{C} f (x, y, z) d s = {鈭�}_{a}^{b} f (x (t), y (t), z (t)) ||r^{'} (t)|| d t$

${鈭�}_{C} (e^{x^{2} + y + z^{2}}) d s = {鈭�}_{0}^{蟺} e^{\cos^{2} t + \sin t + t^{2}} \sqrt{2} d t$

$= \sqrt{2} {鈭�}_{0}^{蟺} e^{\cos^{2} t + \sin t + t^{2}} d t$

$= \sqrt{2} e (e^{n} - 1)$

Hence, ${鈭�}_{C} (e^{x^{2} + y + z^{2}}) d s = \sqrt{2} e (e^{蟺} - 1)$

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