Chapter 5: Q20P (page 241)

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface $z^{2} = 36 x^{2} (4 - x^{2})$

Short Answer

Expert verified

The required solution is 16.

Step by step solution

Definition of double integral

The double integral of f(x,y)over the areaA in the (x,y)plane as the limit of this sum, and we write it as $\int \int_{A} f (x, y) dxdy$

Calculation of the volume under the curve

The volume above the rectangle with vertices at (0,0),(0,1),(2,0), and (2,1), and under the plane z = 8 - x + y .below the surface $z^{2} = 36 x^{2} (4 - x^{2}) .$

First, integrate over z:

$| = \int_{x = 0}^{2} \int_{y = 0}^{1} [\int_{z = 0}^{6 x \sqrt{4 - x^{2}}} d z] d x d y = \int_{0}^{1} d y \int_{0}^{2} d x 6 x \sqrt{4 - x^{2}}$

Further calculation of the volume of the bounded region

Now, integrate over x and y, such that,

$| = \int_{0}^{1} d y \int_{0}^{2} d x 6 x \sqrt{4 - x^{2}} = 6 \int_{0}^{2} \sqrt{4 - x^{2}} d (4 - x^{2}) (- \frac{1}{2}) = {- 3 \frac{2}{3} {(4 - x^{2})}^{3 / 2}}_{0}^{2} = - 2 (0 - 4^{3 / 2)} = 2 \times 2^{3} = 16$

Therefore, the value is 16

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.

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface $z^{2} = 36 x^{2} (4 - x^{2})$

Short Answer

Step by step solution

Definition of double integral

Calculation of the volume under the curve

Further calculation of the volume of the bounded region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Mechanics and Materials

Linear Momentum

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Fields in Physics

Study anywhere. Anytime. Across all devices.

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface z2=36x2(4-x2)

Short Answer

Step by step solution

Definition of double integral

Calculation of the volume under the curve

Further calculation of the volume of the bounded region

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Dynamics

Mechanics and Materials

Linear Momentum

Electric Charge Field and Potential

Magnetism and Electromagnetic Induction

Fields in Physics

Study anywhere. Anytime. Across all devices.

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface $z^{2} = 36 x^{2} (4 - x^{2})$