(8) Please show step by step answer. Thank you in advance. **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.

Let u be the ball.The volume of u is given by the triple integral∫∫∫u1.dV. To compute this, we need…

Answered: Use integration to show that the volume…

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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.

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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(8) Please show step by step answer. Thank you in advance.

**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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