Find volume between given functions about the y = -3 axis. The given image contains two algebraic equations. Here is the transcription:**Equations:**1. $ y = 3(x^2 + 4) $2. $ y = 3(12 - x^2) $**Explanation:**These equations represent quadratic functions. Each equation is in the form of $ y $ as a function of $ x $.1. In the first equation, $ y = 3(x^2 + 4) $, the quadratic term is $ x^2 $ and it has a coefficient of 3. Additionally, there is a constant term, 3 times 4, which adds to the value of y.2. In the second equation, $ y = 3(12 - x^2) $, the quadratic term is also $ x^2 $, but it is subtracted from 12 before being multiplied by 3. This alters the shape and position of the parabolic graph compared to the first equation.Graphs of these equations would display parabolas:- The graph of the first equation, $ y = 3(x^2 + 4) $, is an upward-opening parabola due to the positive coefficient in front of $ x^2 $.- The graph of the second equation, $ y = 3(12 - x^2) $, is a downward-opening parabola because of the negative coefficient when considering the term $ -x^2 $.These equations can be used to understand the effects of different quadratic components and coefficients on the graph of a function.

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Math

Advanced Math

Find volume between given functions about the y = -3 axis.

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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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

Find volume between given functions about the y = -3 axis.

The given image contains two algebraic equations. Here is the transcription:

**Equations:**
1. $ y = 3(x^2 + 4) $
2. $ y = 3(12 - x^2) $

**Explanation:**

These equations represent quadratic functions. Each equation is in the form of $ y $ as a function of $ x $.

1. In the first equation, $ y = 3(x^2 + 4) $, the quadratic term is $ x^2 $ and it has a coefficient of 3. Additionally, there is a constant term, 3 times 4, which adds to the value of y.
2. In the second equation, $ y = 3(12 - x^2) $, the quadratic term is also $ x^2 $, but it is subtracted from 12 before being multiplied by 3. This alters the shape and position of the parabolic graph compared to the first equation.

Graphs of these equations would display parabolas:

- The graph of the first equation, $ y = 3(x^2 + 4) $, is an upward-opening parabola due to the positive coefficient in front of $ x^2 $.
- The graph of the second equation, $ y = 3(12 - x^2) $, is a downward-opening parabola because of the negative coefficient when considering the term $ -x^2 $.

These equations can be used to understand the effects of different quadratic components and coefficients on the graph of a function.

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