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Chapter 14: Q 28. (page 1106)

Evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y, z) = e^{x^{2} + y^{2} + z^{2}}$ , with $C$ the circular helix of radius $1$ , centered about the $y$ -axis, and parametrized by $r (t) = (\cos t, t, \sin t)$ for $蟺鈮� y 鈮� 3 蟺$ .

Short Answer

Expert verified

The required integral is ${鈭�}_{C} (e^{x^{2} + y^{2} + z^{2}}) d s = \sqrt{2} e 蟺^{\frac{3}{2}}$

Step by step solution

01

Given Information

It is given that

$f (x, y, z) = e^{x^{2} + y^{2} + z^{2}}$

Curve is parametrized by

$C : r (t) = 鉄� \cos t, t, \sin t 鉄�$

02

Solving the parametrized curve

The parametric equations are $x (t) = \cos t, y (t) = t, z (t) = \sin t, 蟺鈮� t 鈮� 3 蟺$

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

Also

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

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

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

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

$= \sqrt{2} d t$

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

$鈬� f (x (t), y (t), z (t)) = e^{1 + t^{2}}$

03

Finding the integral

The form of line integral is

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

${鈭�}_{C} (e^{x^{2} + y^{2} + z^{2}}) d s = {鈭�}_{蟺}^{3 蟺} e^{1 + t^{2}} \sqrt{2} d t$

$= \sqrt{2} 路 e {鈭�}_{蟺}^{3 蟺} e^{t^{2}} d t$

$= \sqrt{2} e 路蟺^{\frac{3}{2}}$

Hence, ${鈭�}_{C} (e^{x^{2} + y^{2} + z^{2}}) d s = \sqrt{2} e 蟺^{\frac{3}{2}}$

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Most popular questions from this chapter

Q. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A smooth surface with a smooth boundary.

(b) A surface that is not smooth, but that has a smooth boundary.

(c) A surface that is smooth, but does not have a smooth boundary

Suppose that an electric field is given by

$E = 2 y i + 2 x y j + y z k$

Compute the flux ${鈭�}_{S} 鈥� E 鈰� n d A$ of the field through the unit cube $[0, 1] 脳 [0, 1] 脳 [0, 1]$ .

Find

$\begin{aligned} {鈭�}_{S} 鈥� F (x, y, z) 鈰� n d S if \\ F (x, y, z) = 2 x z i + 2 y z j 鈭� 18 k \end{aligned}$

and S is the portion of the hyperboloid $x^{2} + y^{2} - 9 = z^{2}$ that lies between the planes

z = 鈭�4 and z = 0, with n pointing outwards.

Consider the vector field $F (x, y, z) = (y z, x z, x y)$ . Find a vector field $G (x, y, z)$ with the property that, for all points in role="math" localid="1650383268941" ${鈩�}^{3}, G (x, y, z) = 2 F (x, y, z)$ .

Use the curl form of Green鈥檚 Theorem to write the line integral of F(x, y) about the unit circle as a double integral. Do not evaluate the integral.

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