Chapter 13: Q. 1 (page 1041)

Let be real $α, β, γ, δ, ϵ, and ζ$ numbers. Evaluate the triple iterated integral
$\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x$
What does this integral represent?

Short Answer

Expert verified

Integral is evaluated as $\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x = (ζ - ϵ) (δ - γ) (β - α)$ and represents volume.

Step by step solution

Step 1. Given information

The given integral is $\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x$

Step 2. Simplification

Evaluate the integral

$\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x = \int_{α}^{β} (\int_{γ}^{δ} (\int_{ϵ}^{ζ} 1 d z) d y) d x \int_{α}^{β} (\int_{γ}^{δ} ({[z]}_{ϵ}^{ζ}) d y) d x = \int_{α}^{β} (\int_{γ}^{δ} (ζ - ϵ) d y) d x = \int_{α}^{β} ({[ζ - ϵ]}_{γ}^{δ}) d x = \int_{α}^{β} ((ζ - ϵ) (δ - γ)) d x = {[(ζ - ϵ) (δ - γ)]}_{α}^{β} = (ζ - ϵ) (δ - γ) (β - α)$

So integral is evaluated as $\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x = (ζ - ϵ) (δ - γ) (β - α) .$

The triple-iterated integral represents volume.

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Let be real $α, β, γ, δ, ϵ, and ζ$ numbers. Evaluate the triple iterated integral
$\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x$
What does this integral represent?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Simplification

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Let be real α,β,γ,δ,ϵ,andζnumbers. Evaluate the triple iterated integral∫αβ∫γδ∫ϵζdzdydxWhat does this integral represent?

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

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Study anywhere. Anytime. Across all devices.

Let be real $α, β, γ, δ, ϵ, and ζ$ numbers. Evaluate the triple iterated integral
$\int_{α}^{β} \int_{γ}^{δ} \int_{ϵ}^{ζ} d z d y d x$
What does this integral represent?