Q. 31 Integrate the given function ove... [FREE SOLUTION]

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Chapter 14: Q. 31 (page 1119)

Integrate the given function over the accompanying surface in Exercises 27鈥�34.
$f (x, y, z) = \frac{y}{x} \sqrt{4 z^{2} + 1}$ , where S is the portion of the paraboloid $z = x^{2} + y^{2}$ that lies above the rectangle determined by $1 鈮� x 鈮� e$ and $0 鈮� y 鈮� 2$ in the xyplane.

Short Answer

Expert verified

The integration of the function $f (x, y, z) = \frac{y}{x} \sqrt{4 z + 1}$ is $4 e^{2} + 14$ .

Step by step solution

Step 1. Given Information.

The function:

$f (x, y, z) = \frac{y}{x} \sqrt{4 z + 1}$

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 follows:

$r (u, v) = (u, v, u^{2} + v^{2}) r_{u} = \frac{鈭� r}{鈭� u} = (1, 0, 2 u) r_{v} = \frac{鈭� r}{鈭� v} = (0, 1, 2 v)$

Step 4. Find ru×rv

$r_{u} 脳 r_{v} = (1, 0, 2 u) 脳 (0, 1, 2 v) = - 2 u i - 2 v j + 1 k ||r_{u} 脳 r_{v}|| = \sqrt{{(- 2 u)}^{2} + {(- 2 v)}^{2} + 1^{2}} = \sqrt{4 u^{2} + 4 v^{2} + 1}$

Step 5. Find the parametrisation function.

$f (x, y, z) = \frac{y}{z} \sqrt{4 z + 1} f (x (u, v), y (u, v), z (u, v)) = \frac{v}{u} \sqrt{4 (u^{2} + v^{2}) + 1} = \frac{v}{u} \sqrt{4 u^{2} + 4 v^{2} + 1}$

Step 6. Find the region of integration and integrate.

The region of integration is given by,

$D = {(u, v) 1 鈮� u 鈮� e, 0 鈮� v 鈮� 2} {鈭�}_{S} d S = 鈭� {鈭�}_{D} f (x (u, v), z (u, v)) ||r_{u} 脳 r_{v}|| d u d v = {鈭�}_{0}^{2} {鈭�}_{1}^{e} \frac{v}{u} (4 u^{2} + 4 v^{2} + 1) d u d v = {鈭�}_{0}^{2} {鈭�}_{1}^{e} (4 u v + \frac{4 v^{3}}{u} + \frac{v}{u}) d u d v = {鈭�}_{0}^{2} {[2 u^{2} v + 4 v^{3} \ln u + v \ln u]}_{1}^{e} d v = {鈭�}_{0}^{2} [2 e^{2} v + 4 v^{3} - v] d v = {[e^{2} v^{2} + v^{4} - \frac{v^{2}}{2}]}_{0}^{2} = 4 e^{2} + 14$

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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)=yx4z2+1, where S is the portion of the paraboloid z=x2+y2that lies above the rectangle determined by 1鈮�x鈮�eand 0鈮�y鈮�2in the xyplane.

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

Applied Mathematics

Statistics

Pure Maths

Discrete Mathematics

Decision Maths

Calculus

Study anywhere. Anytime. Across all devices.