Chapter 5: Q11P (page 268)

Write a triple integral in cylindrical coordinates for the volume inside the cylinder $x^{2} + y^{2} = 4$ and between $z = 2 x^{2} + y^{2}$ and the (x,y) plane. Evaluate the integral.

Short Answer

Expert verified

The required volume is $12 τ τ$

Step by step solution

Given information

The solid is bounded by the cylinder the cylinder $x^{2} + y^{2} = 4$ and between $z = 2 x^{2} + y^{2}$ and the (x,y) plane.

Concept of spherical coordinates

The spherical coordinates $(r, θ, ϕ)$ is related to Cartesian coordinates (x,y,z) by:

$r = \sqrt{x^{2} + y^{2} + z^{2} θ} = \tan^{- 1} (\frac{y}{x}) ϕ = \cos^{- 1} (\frac{z}{r})$

The cylindrical coordinates $(r, θ, ϕ)$ is related to Cartesian coordinates(x,y,z) by:

$r = \sqrt{x^{2} + y^{2} θ} = t a n^{- 1} (\frac{y}{x}) z = z$

The limits of polar coordinates

The required integral is I $\int_{0}^{2} d x \int_{0}^{\sqrt{4 - x^{2}}} d y \int_{0}^{2 x^{2} + y^{2}} d z$ .

$r : 0 \to 2 θ : 0 \to 2 τ τ z : 0 \to 2 r^{2} \cos^{2} θ + r^{2} (\cos^{2} θ + 1)$

Calculate further as follows:

$I = \int_{0}^{2} d x \int_{0}^{\sqrt{4 - x^{2}}} d y \int_{0}^{2 x^{2} + y^{2}} d z = \int_{0}^{2} r d r \int_{0}^{2 τ τ} d θ \int_{0}^{r^{2} (\cos^{2} θ + 1)} d z = \int_{0}^{2} r d r \int_{0}^{2 τ τ} r^{2} (\cos^{2} θ + 1) d θ$

The above integral can be written as follows:

$I = \int_{0}^{2} r^{3} d r \int_{0}^{2 τ τ} r^{2} (\cos^{2} θ + 1) d θ = {(\frac{r^{4}}{4})}_{0}^{2} (2 τ τ + \frac{1}{2} \int_{0}^{2} (\cos 2 θ + 1) d θ) = 4 (2 τ τ + τ τ) = 12 τ τ$

Therefore, the required volume is $12 τ τ$ .

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.

Write a triple integral in cylindrical coordinates for the volume inside the cylinder $x^{2} + y^{2} = 4$ and between $z = 2 x^{2} + y^{2}$ and the (x,y) plane. Evaluate the integral.

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

The limits of polar coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Particle Model of Matter

Geometrical and Physical Optics

Electric Charge Field and Potential

Work Energy and Power

Kinematics Physics

Study anywhere. Anytime. Across all devices.

Write a triple integral in cylindrical coordinates for the volume inside the cylinder x2+y2=4and between z=2x2+y2 and the (x,y) plane. Evaluate the integral.

Short Answer

Step by step solution

Given information

Concept of spherical coordinates

The limits of polar coordinates

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Turning Points in Physics

Particle Model of Matter

Geometrical and Physical Optics

Electric Charge Field and Potential

Work Energy and Power

Kinematics Physics

Study anywhere. Anytime. Across all devices.

Write a triple integral in cylindrical coordinates for the volume inside the cylinder $x^{2} + y^{2} = 4$ and between $z = 2 x^{2} + y^{2}$ and the (x,y) plane. Evaluate the integral.