Chapter 14: Q. 31 (page 1119)

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

$\int_{S} d S = \int \int_{D} f (x (u, v), y (u, v), z (u, v)) ||r_{u} \times r_{v}|| d u d v$

Step 3. Find ru,rv

The surface is parametrized as follows:

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

Step 4. Find ru×rv

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

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,

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

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

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Integrate the given function over the accompanying surface in Exercises 27–34.f(x,y,z)=yx4z2+1, where S is the portion of the paraboloid z=x2+y2that lies above the rectangle determined by 1≤x≤eand 0≤y≤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

Theoretical and Mathematical Physics

Probability and Statistics

Mechanics Maths

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Geometry

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Study anywhere. Anytime. Across all devices.

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