Chapter 5: Q43P (page 248)

Find the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0), (1,0) and (1,2).

Short Answer

Expert verified

The volume obtained for the planes over the vertices is $\frac{9}{2}$

Step by step solution

Definition of double integral and mass formula

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

Draw the area bounded by the curve in the plane

Draw the area bounded by region for the integration in the xy plane.

Calculation of the volume under the curve

Calculate the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0) , (1,0) and (1,2).

$l = \int_{0}^{1} d x \int_{0}^{2 x} d y \int_{x^{2} + y^{2} + 8}^{2 x^{2} + y^{2} + 12} d z = \int_{0}^{1} d x \int_{0}^{2 x} (- x^{2} - y^{2} - 8 + 2 x^{2} + y^{2} + 12) d y = \int_{0}^{1} d x \int_{0}^{2 x} (x^{2} + 4) d y = \int_{0}^{1} d x (x^{2} + 4) 2 x$

Simplify and solve further.

$\int_{0}^{1} d x (x^{2} + 4) 2 x = \int_{0}^{1} (2 x^{3} + 8 x) d x = {[\frac{1}{2} x^{4} + 4 x^{2}]}_{0}^{1} = (\frac{1}{2} + 4) = \frac{9}{2}$

Therefore, the value is $\frac{9}{2}$ ,

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.

Find the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0), (1,0) and (1,2).

Short Answer

Step by step solution

Definition of double integral and mass formula

Draw the area bounded by the curve in the plane

Calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Measurements

Magnetism

Space Physics

Translational Dynamics

Wave Optics

Study anywhere. Anytime. Across all devices.

Find the volume between the surfaces z=2x2+y2+12and z=x2+y2+8, and over the triangle with vertices (0,0), (1,0) and (1,2).

Short Answer

Step by step solution

Definition of double integral and mass formula

Draw the area bounded by the curve in the plane

Calculation of the volume under the curve

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Measurements

Magnetism

Space Physics

Translational Dynamics

Wave Optics

Study anywhere. Anytime. Across all devices.

Find the volume between the surfaces $z = 2 x^{2} + y^{2} + 12$ and $z = x^{2} + y^{2} + 8$ , and over the triangle with vertices (0,0), (1,0) and (1,2).