Q. 21 聽Find the area of S is the port... [FREE SOLUTION]

91影视

Chapter 14: Q. 21 (page 1119)

Find the area of S is the portion of the plane with equation y鈭抸 = $\frac{蟺}{2}$
that lies above the rectangle determined by 0 鈮� x 鈮� 4 and 3 鈮� y 鈮� 6.

Short Answer

Expert verified

$Hence the area of triangle is = 12 \sqrt{2} .$

Step by step solution

Step 1. Given

Equation of plane is y鈭抸 = $\frac{蟺}{2}$

Step 2. Formula for finding the area of surface

$If a surface S is given by z = f (x, y) for f (x, y) 鈭� D 鈯� R^{2} Then the surface area of smooth surface is \begin{aligned} {鈭�}_{S} 鈥� d S = {鈭�}_{S} 鈥� \sqrt{{(\frac{鈭� z}{鈭� x})}^{2} + {(\frac{鈭� z}{鈭� y})}^{2} + 1 d A} \\ = {鈭�}_{D} 鈥� \sqrt{{(\frac{鈭� z}{鈭� x})}^{2} + {(\frac{鈭� z}{鈭� y})}^{2} + 1 d A 鈰� 鈰� (1} \end{aligned}$

Step 3. Finding the partial derivative

$y - z = \frac{蟺}{2} t h e n, z = y - \frac{蟺}{2} n o w, f i r s t f i n d \begin{aligned} \frac{鈭� z}{鈭� x} = \frac{鈭�}{鈭� x} (y 鈭� \frac{蟺}{2}) \\ = 0 \end{aligned} a n d \begin{aligned} \frac{鈭� z}{鈭� y} = \frac{鈭�}{鈭� y} (y 鈭� \frac{蟺}{2}) \\ = 1 \end{aligned}$

Step 4. Finding area

$T h e a r e a o f s u r f a c e i s \begin{aligned} {鈭�}_{S} 鈥� d S = {鈭�}_{D} 鈥� \sqrt{{(\frac{鈭� z}{鈭� x})}^{2} + {(\frac{鈭� z}{鈭� y})}^{2} + 1} d A \\ = {鈭�}_{3}^{6} 鈥� {鈭�}_{0}^{4} 鈥� \sqrt{0^{2} + 1^{2} + 1} d x d y \\ = {鈭�}_{3}^{6} 鈥� {鈭�}_{0}^{4} 鈥� \sqrt{2} d x d y \\ = \sqrt{2} {鈭�}_{3}^{6} 鈥� {鈭�}_{0}^{4} 鈥� d x d y \\ = \sqrt{2} {鈭�}_{3}^{6} 鈥� [{鈭�}_{0}^{4} 鈥� d x] d y \\ = \sqrt{2} {鈭�}_{3}^{6} 鈥� [x]_{0}^{4} d y \\ = \sqrt{2} {鈭�}_{3}^{6} 鈥� [4 鈭� 0] d y \\ = \sqrt{2} {鈭�}_{3}^{6} 鈥� 4 d y \\ = 4 \sqrt{2} {鈭�}_{3}^{6} 鈥� d y \\ = 4 \sqrt{2} [y]_{3}^{6} \\ = 4 \sqrt{2} [6 鈭� 3] \\ = 4 \sqrt{2} [3] \\ = 12 \sqrt{2} . \\ Hence the area of triangle is = 12 \sqrt{2} . \end{aligned}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the area of S is the portion of the plane with equation y鈭抸 = $\frac{蟺}{2}$
that lies above the rectangle determined by 0 鈮� x 鈮� 4 and 3 鈮� y 鈮� 6.

Short Answer

Step by step solution

Step 1. Given

Step 2. Formula for finding the area of surface

Step 3. Finding the partial derivative

Step 4. Finding area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Geometry

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

Find the area of S is the portion of the plane with equation y鈭抸 =蟺2that lies above the rectangle determined by 0 鈮� x 鈮� 4 and 3 鈮� y 鈮� 6.

Short Answer

Step by step solution

Step 1. Given

Step 2. Formula for finding the area of surface

Step 3. Finding the partial derivative

Step 4. Finding area

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Geometry

Applied Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the area of S is the portion of the plane with equation y鈭抸 = $\frac{蟺}{2}$
that lies above the rectangle determined by 0 鈮� x 鈮� 4 and 3 鈮� y 鈮� 6.