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 $12 τ τ$ .

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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

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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

Linear Momentum

Solid State Physics

Waves Physics

Translational Dynamics

Modern Physics

Turning Points in 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.