Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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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Transcribed Image Text: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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