Chapter 5: Q16P (page 247)

$\iint {(9 + 2 y^{2})}^{- 1}$ over the quadrilateral with vertice $(1, 3), (3, 3), (2, 6), (6, 6)$

Short Answer

Expert verified

The required solution is $\frac{In 3}{6}$

Step by step solution

Definition of double integral

The double integral of $f (x, y)$ over the areain the $(x, y)$ plane as the limit of this sum, and we write it as $\iint_{A} f (x, y) d x d y$ .

Step 2: Drawing the area bounded by the curve

The area is bounded by the quadrilateral with vertices $(1, 3), (3, 3), (2, 6), (6, 6)$ .

Calculation of the area under the curve

First, integrate over x:

$\iint_{A} \frac{d y d x}{9 + 2 y^{2}} = \int_{y = 3}^{6} [\int_{x = y / 3}^{y} d x] \frac{d y}{9 + 2 y^{2}} = \int_{3}^{6} \frac{2}{3} y \frac{d y}{2 y^{2}}$

Further calculation of the area of the bounded region

Now, integrate over y:

$\frac{2}{3} \int_{3}^{6} \frac{y d y}{9 + 2 y^{2}} = \frac{1}{6} \int_{3}^{6} \frac{(d 9 + 2 y^{2})}{9 + 2 y^{2}} = \frac{1}{6} I n {(9 + 2 y^{2})}_{3}^{6} = \frac{I n 3}{6}$

Therefore, the value is $\frac{I n 3}{6}$

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

$\iint {(9 + 2 y^{2})}^{- 1}$ over the quadrilateral with vertice $(1, 3), (3, 3), (2, 6), (6, 6)$

Short Answer

Step by step solution

Definition of double integral

Step 2: Drawing the area bounded by the curve

Calculation of the area under the curve

Further calculation of the area of the bounded region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Energy Physics

Electromagnetism

Electric Charge Field and Potential

Waves Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

∬(9+2y2)-1 over the quadrilateral with vertice(1,3),(3,3),(2,6),(6,6)

Short Answer

Step by step solution

Definition of double integral

Step 2: Drawing the area bounded by the curve

Calculation of the area under the curve

Further calculation of the area of the bounded region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Scientific Method Physics

Energy Physics

Electromagnetism

Electric Charge Field and Potential

Waves Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

$\iint {(9 + 2 y^{2})}^{- 1}$ over the quadrilateral with vertice $(1, 3), (3, 3), (2, 6), (6, 6)$