Chapter 14: Q 21. (page 1106)

Evaluate the multivariate line integral of the given function over the specified curve.
$g (x, y) = x$ with $C$ , the graph of $y = x^{2}$ in $x y$ plane from origin to $(2.4)$

Short Answer

Expert verified

The required value is $\frac{1}{12} (17^{3 / 2} - 1)$ .

Step by step solution

Given Information

It is given that $g (x, y) = x$

The curve $C$ is the graph of $y = x^{2}$ in the $x y$ -plane from the origin to $(2, 4)$

Formula for line integral

It is given by

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

Solve the parametrization curve

Parametrization curve be chosen as $x = t, y = t^{2}, 0 \leq t \leq 2$

$\Rightarrow x^{'} (t) = 1, y^{'} (t) = 2 t$

Also

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

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

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

$= \sqrt{1 + 4 t^{2}} d t$

Therefore

$\int_{C} g (x, y) d s = \int_{0}^{2} t \sqrt{1 + 4 t^{2}} d t$

Assuming $u = 1 + 4 t^{2}$

$\Rightarrow d u = 8 t d t$

$\Rightarrow \frac{d u}{8} = t d t$

Finding line integral

Limits are as below:

$t = 0 then u = 1$

$t = 2 then u = 17$

Therefore

$\int_{C} g (x, y) d s = \int_{0}^{2} t \sqrt{1 + 4 t^{2}} d t$

$= \int_{0}^{2} \sqrt{1 + 4 t^{2}} t d t$

$= \int_{1}^{17} \sqrt{u} \frac{d u}{8}$

$= \frac{1}{8} \int_{1}^{17} \sqrt{u} d u$

$= \frac{1}{8} (\frac{2}{3}) {(u^{3 / 2})}_{1}^{17}$

$= \frac{1}{12} (17^{3 / 2} - 1)$

Hence, required integral is $\frac{1}{12} (17^{3 / 2} - 1)$

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Evaluate the multivariate line integral of the given function over the specified curve.
$g (x, y) = x$ with $C$ , the graph of $y = x^{2}$ in $x y$ plane from origin to $(2.4)$

Short Answer

Step by step solution

Given Information

Formula for line integral

Solve the parametrization curve

Finding line integral

One App. One Place for Learning.

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Evaluate the multivariate line integral of the given function over the specified curve.g(x,y)=xwithC, the graph ofy=x2inxyplane from origin to2.4

Short Answer

Step by step solution

Given Information

Formula for line integral

Solve the parametrization curve

Finding line integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Statistics

Mechanics Maths

Logic and Functions

Decision Maths

Study anywhere. Anytime. Across all devices.

Evaluate the multivariate line integral of the given function over the specified curve.
$g (x, y) = x$ with $C$ , the graph of $y = x^{2}$ in $x y$ plane from origin to $(2.4)$