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 $- 1 \leq y \leq 1$

Short Answer

Expert verified

The required value is $26.7036$

Step by step solution

Given Information

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

Curve $C$ is line with equation $x = 3 y$ for $- 1 \leq y \leq 1$ .

Formula for line integral

It is given by

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

Solving the parametrization curve

Selecting the curve be $x = 3 t, y = t, - 1 \leq t \leq 1 .$

$\Rightarrow x^{'} (t) = 3, y^{'} (t) = 1$

Also

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

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

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

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

$= \sqrt{10} d t$

Finding the line integral

Solving further, we get

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

$= \int_{- 1}^{1} e^{3 t^{2}} \sqrt{10} d t$

$= \sqrt{10} \int_{- 1}^{3 t^{2}} d t$

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

$= \sqrt{10} (8.44442)$

$\approx 26.7036$

Hence, the value is $\approx 26.7036$ (Approx.)

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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 $- 1 \leq y \leq 1$

Short Answer

Step by step solution

Given Information

Formula for line integral

Solving the parametrization curve

Finding the line integral

One App. One Place for Learning.

Most popular questions from this chapter

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Evaluate the multivariate line integral of the given function over the specified curve.f(x,y)=exywithCthe line with equationx=3yfor-1≤y≤1

Short Answer

Step by step solution

Given Information

Formula for line integral

Solving the parametrization curve

Finding the line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Discrete Mathematics

Probability and Statistics

Statistics

Study anywhere. Anytime. Across all devices.

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 $- 1 \leq y \leq 1$