**Educational Transcription: Calculating the Volume of a Solid of Revolution****Problem Statement:**Find the volume of the solid generated by revolving the shaded region about the given axis.**Details:**- **Equation:** $ x = \frac{y^2}{5} $- **Axis of Revolution:** About the y-axis**Graph Description:**- The graph displays a shaded region bounded by the curve $ x = \frac{y^2}{5} $, the y-axis, and horizontal lines at $ y = 1 $ and $ y = 6 $.- The region is a section of the xy-plane that resembles a parabolic shape extending from $ y = 1 $ to $ y = 6 $.**Volume Calculation Options:**1. $ \frac{625}{6} \pi $2. $ 25\pi $3. $ \frac{250}{3} \pi $4. $ 100\pi $**Instructions:**- Click "Save and Submit" to save and submit your answer.- Click "Save All Answers" to save all your answers.For an educational context, remember to understand the process of revolving a region to find the volume, using integration techniques such as the disk or washer methods, especially when dealing with functions in terms of $ y $.

When x=5 then x=y255=y25y2=25y=5 So y is from 0 to 5 in the figure.

Answered: Find the volume of the solid generated…

Find the volume of the solid generated by revolving the shaded region about the given axis. About the y- axis 5- 4+ 3+ 2+ 1 4. 5 625 TU 25 250 100

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.3: Hyperbolas

Problem 65E

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Concept explainers

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.

Question

**Educational Transcription: Calculating the Volume of a Solid of Revolution**

**Problem Statement:**
Find the volume of the solid generated by revolving the shaded region about the given axis.

**Details:**
- **Equation:** $ x = \frac{y^2}{5} $
- **Axis of Revolution:** About the y-axis

**Graph Description:**
- The graph displays a shaded region bounded by the curve $ x = \frac{y^2}{5} $, the y-axis, and horizontal lines at $ y = 1 $ and $ y = 6 $.
- The region is a section of the xy-plane that resembles a parabolic shape extending from $ y = 1 $ to $ y = 6 $.

**Volume Calculation Options:**
1. $ \frac{625}{6} \pi $
2. $ 25\pi $
3. $ \frac{250}{3} \pi $
4. $ 100\pi $

**Instructions:**
- Click "Save and Submit" to save and submit your answer.
- Click "Save All Answers" to save all your answers.

For an educational context, remember to understand the process of revolving a region to find the volume, using integration techniques such as the disk or washer methods, especially when dealing with functions in terms of $ y $.

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