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Chapter 14: Q 22. (page 1106)

Evaluate the multivariate line integral of the given function over the specified curve.
$f (x, y) = e^{x y}$ with $C$ the line with equation $x = 3 y$ for $- 1 鈮� y 鈮� 1$

Short Answer

Expert verified

The required value is $26.7036$

Step by step solution

01

Given Information

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

Curve $C$ is line with equation $x = 3 y$ for $- 1 鈮� y 鈮� 1$ .

02

Formula for line integral

It is given by

${鈭�}_{C} f (x, y) d s = {鈭�}_{a}^{b} f (x (t), y (t)) \sqrt{{[x^{'} (t)]}^{2} + {[y^{'} (t)]}^{2}} d t$

03

Solving the parametrization curve

Selecting the curve be $x = 3 t, y = t, - 1 鈮� t 鈮� 1 .$

$鈬� x^{'} (t) = 3, y^{'} (t) = 1$

Also

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

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

$= \sqrt{(3)^{2} + (1)^{2}} d t$

$= \sqrt{9 + 1} d t$

$= \sqrt{10} d t$

04

Finding the line integral

Solving further, we get

${鈭�}_{C} f (x, y) d s = {鈭�}_{- 1}^{1} e^{(3 t) (t)} \sqrt{10} d t$

$= {鈭�}_{- 1}^{1} e^{3 t^{2}} \sqrt{10} d t$

$= \sqrt{10} {鈭�}_{- 1}^{3 t^{2}} d t$

$= \sqrt{10} (\sqrt{\frac{蟺}{3}} erfi (\sqrt{3}))$

$= \sqrt{10} (8.44442)$

$鈮� 26.7036$

Hence, the value is $鈮� 26.7036$ (Approx.)

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Most popular questions from this chapter

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.

$f (x, y, z) = <\frac{1}{x \ln (z + 2 y)}>$ , where Sis the surface given by $r (u, v) = 2 u i + (u + v) j + (2 u - 2 v) k$ for $3 鈮� u 鈮� 7$ and $2 鈮� v 鈮� 10$ .

$F (x, y, z) = 鈭� y i + x j 鈭� e^{y z} k$ , where S is the cylinder with equation $x^{2} + y^{2} = 9$ from $z = 2, 4$ , with n pointing outwards.

Compute dS for your parametrization in Exercise 7.

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" ${鈩�}^{3}, G (x, y, z) = 2 F (x, y, z)$ .

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