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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

01

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 $(2, 4)$

02

Formula for line integral

It is given by

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

03

Solve the parametrization curve

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

$鈬� x^{'} (t) = 1, y^{'} (t) = 2 t$

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

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

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

$鈬� d u = 8 t d t$

$鈬� \frac{d u}{8} = t d t$

04

Finding line integral

Limits are as below:

$t = 0 then u = 1$

$t = 2 then u = 17$

Therefore

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

$= {鈭�}_{0}^{2} \sqrt{1 + 4 t^{2}} t d t$

$= {鈭�}_{1}^{17} \sqrt{u} \frac{d u}{8}$

$= \frac{1}{8} {鈭�}_{1}^{17} \sqrt{u} d u$

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

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

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

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Most popular questions from this chapter

Find ${鈭�}_{S} 鈥� 1 d S$ , where S is the portion of the surface determined by $z = x^{2} 鈭� \sqrt{3} y$ that lies above the region in the xy-plane bounded by the x-axis and the lines with equations $y = 2 x, x = 3$ .

Find the flux of the given vector field through a permeable membrane described by surface S.

$F (x, y, z) = y i + x j + k$ , where S is the paraboloid with equation role="math" localid="1650299986090" $z = 4 x^{2} + y^{2} + 1$ that lies above the annulus determined by $1 鈮� x^{2} + y^{2} 鈮� 4$ in the XY-plane.

$F (x, y, z) = 鈭� x z i 鈭� y z j + z^{2} k$ , where S is the cone with equation $z = \sqrt{x^{2} + y^{2}}$ between $z = 2, 4$ , with n pointing outwards.

Compute n for the surface S in Exercise 12.

Find the masses of the lamina:

The lamina occupies the region of the hyperboloid with equation $z^{2} + 4 = x^{2} + y^{2}$ that lies above and/or below the disk of radius 5 about the origin in the XY-plane, and the density function, 蟻(x, y,z), is proportional to the distance from the origin.

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