Chapter 13: Q 44 (page 1067)

The iterated integrals use spherical coordinates. Describe the solids determined by the limits of integration.
$\int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} f (ρ, θ, ϕ) ρ^{2} \sin ϕ d ρ d θ d ϕ$

Short Answer

Expert verified

The region represents sphere with second quadrant with the hemisphere below the $x y -$ plane of the sphere with radius 1 centered at origin

Step by step solution

Step 1:Given information

The given expression is $\int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} f (ρ, θ, ϕ) ρ^{2} \sin ϕ d ρ d θ d ϕ$

Step 2:Simplification

$\int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} f (ρ, θ, ϕ) ρ^{2} \sin ϕ d ρ d θ d ϕ$

According to the order of integration

$\Rightarrow ρ = 0 t o ρ = 1$

$\Rightarrow \sqrt{x^{2} + y^{2} + z^{2}} = 1$

$\Rightarrow x^{2} + y^{2} + z^{2} = 1^{2}$

This is a sphere equation with radius 1 centered at origin

Now,

$\Rightarrow θ$ varies from $0 t o \frac{π}{2}$

It means it is in second quadrant

Now,

$\Rightarrow ϑ$ varies from $0 t o \frac{π}{2}$

these the region represents the hemisphere below the xy -plane of the sphere with radius 1 centered at origin

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The iterated integrals use spherical coordinates. Describe the solids determined by the limits of integration.
$\int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} f (ρ, θ, ϕ) ρ^{2} \sin ϕ d ρ d θ d ϕ$

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

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The iterated integrals use spherical coordinates. Describe the solids determined by the limits of integration.∫0π2∫0π2∫01fρ,θ,ϕρ2sinϕdρdθdϕ

Short Answer

Step by step solution

Step 1:Given information

Step 2:Simplification

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Theoretical and Mathematical Physics

Discrete Mathematics

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

The iterated integrals use spherical coordinates. Describe the solids determined by the limits of integration.
$\int_{0}^{\frac{π}{2}} \int_{0}^{\frac{π}{2}} \int_{0}^{1} f (ρ, θ, ϕ) ρ^{2} \sin ϕ d ρ d θ d ϕ$