Q. 28 Integrate the given function ove... [FREE SOLUTION]

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Chapter 14: Q. 28 (page 1119)

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
$f (x, y, z) = x^{2} - y + 3 z$ , where Sis the portion of the plane with equation $2 x - 6 y + 3 z = 1$ whose preimage in the xz plane is the region bounded by the coordinate axes and the lines with equations z = 4 and x = z.

Short Answer

Expert verified

The integration of the function $f (x, y, z) = x^{2} - y + 3 z$ is $\frac{2282}{27}$ .

Step by step solution

Step 1. Given Information.

The function:

$f (x, y, z) = x^{2} - y + 3 z 2 x - 6 y + 3 z = 1$

Step 2. Formula for finding the integral.

The formula for finding the integral over the smooth surface is given by,

${鈭�}_{S} d S = 鈭� {鈭�}_{D} f (x (u, v), y (u, v), z (u, v)) ||r_{u} 脳 r_{v}|| d u d v$

Step 3. Find ru,rv.

The surface is parametrized as:
$r (u, v) = (u, - \frac{1}{6} (1 - 2 u - 3 v), v) r_{u} = \frac{鈭� r}{鈭� u} = (1, \frac{1}{3}, 0) r_{v} = \frac{鈭� r}{鈭� v} = (0, \frac{1}{2}, 1)$

Step 4. Find ru×rv

$r_{u} 脳 r_{v} = (1, \frac{1}{3}, 0) 脳 (0, \frac{1}{2}, 1) = \frac{1}{3} i - j + \frac{1}{2} k ||r_{u} 脳 r_{v}|| = \sqrt{{(\frac{1}{3})}^{2} + {(- 1)}^{2} + {(\frac{1}{2})}^{2}} = \sqrt{\frac{49}{36}} = \frac{7}{6}$

Step 5. Find the parametrisation function.

$f (x, y, z) = x^{2} - y + 3 z f (x (u, v), y (u, v), z (u, v)) = u^{2} - (\frac{1}{6} (1 - 2 u - 3 v)) + 3 v = u^{2} - \frac{1}{3} u + \frac{5}{2} v + \frac{1}{6}$

Step 6. Find the region of integration and integrate.

The region of integration is given by,

$D = {(u, v) 0 鈮� u 鈮� v, 0 鈮� v 鈮� 4} {鈭�}_{S} d S = 鈭� {鈭�}_{D} f (x (u, v), z (u, v)) ||r_{u} 脳 r_{v}|| d u d v = {鈭�}_{0}^{4} {鈭�}_{0}^{v} [u^{2} - \frac{1}{3} u + \frac{5}{2} v + \frac{1}{6}] \frac{7}{6} d u d v = \frac{7}{6} {鈭�}_{0}^{4} {[\frac{u^{3}}{3} - \frac{u^{2}}{6} + \frac{5}{2} u v + \frac{1}{6} u]}_{0}^{v} = \frac{7}{6} {鈭�}_{0}^{4} [\frac{v^{3}}{3} + \frac{14 v^{2}}{6} + \frac{v}{6}] d v = \frac{7}{6} {[\frac{v^{4}}{12} + \frac{14 v^{2}}{18} + \frac{v^{2}}{12}]}_{0}^{4} = \frac{7}{6} (\frac{652}{9}) = \frac{2282}{27}$

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Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formula for finding the integral.

Step 3. Find ru,rv.

Step 4. Find ru×rv

Step 5. Find the parametrisation function.

Step 6. Find the region of integration and integrate.

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91影视

Integrate the given function over the accompanying surface in Exercises 27鈥�34.f(x,y,z)=x2-y+3z, where Sis the portion of the plane with equation 2x-6y+3z=1whose preimage in the xz plane is the region bounded by the coordinate axes and the lines with equations z = 4 and x = z.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Formula for finding the integral.

Step 3. Find ru,rv.

Step 4. Find ru×rv

Step 5. Find the parametrisation function.

Step 6. Find the region of integration and integrate.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Statistics

Geometry

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.