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Use integration to show that the volume of the closed unit ball B = {(x, y, z) : x2 + y? + 22 < 1} in R³ is 47/3.

Use integration to show that the volume of the closed unit ball B = {(x, y, z) : x2 + y? + 22 < 1} in R³ is 47/3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Volume of a Closed Unit Ball Using Integration** Use integration to show that the volume of the closed unit ball \( B = \{ (x, y, z) : x^2 + y^2 + z^2 \leq 1 \} \) in \( \mathbb{R}^3 \) is \(\frac{4\pi}{3}\). **Explanation:** The problem asks for the calculation of the volume of a 3-dimensional unit sphere (or "ball") using integration. The unit ball in this context is defined by the set of points \((x, y, z)\) that satisfy the inequality \(x^2 + y^2 + z^2 \leq 1\). The task is to use calculus to prove that the resulting volume is \(\frac{4\pi}{3}\). To achieve this, one typically uses spherical coordinates or evaluates a triple integral in Cartesian coordinates. The integration process involves calculating the volume element and integrating over the sphere's entire volume.
Transcribed Image Text:**Title: Volume of a Closed Unit Ball Using Integration** Use integration to show that the volume of the closed unit ball \( B = \{ (x, y, z) : x^2 + y^2 + z^2 \leq 1 \} \) in \( \mathbb{R}^3 \) is \(\frac{4\pi}{3}\). **Explanation:** The problem asks for the calculation of the volume of a 3-dimensional unit sphere (or "ball") using integration. The unit ball in this context is defined by the set of points \((x, y, z)\) that satisfy the inequality \(x^2 + y^2 + z^2 \leq 1\). The task is to use calculus to prove that the resulting volume is \(\frac{4\pi}{3}\). To achieve this, one typically uses spherical coordinates or evaluates a triple integral in Cartesian coordinates. The integration process involves calculating the volume element and integrating over the sphere's entire volume.
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