Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y = 7x² and y = x² + 8. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?
Farmer Jones, and his wife, Dr. Jones, decide to build a fence in their field, to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by y = 7x² and y = x² + 8. Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife. What is the area of the enclosed region?
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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![**Mathematical Enclosures for Farm Fields**
Farmer Jones and his wife, Dr. Jones, decide to build a fence in their field to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by the equations \( y = 7x^2 \) and \( y = x^2 + 8 \). Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife.
**Problem:**
Given the equations of these curves, determine the area of the enclosed region.
**Explanation:**
1. **Equations of the Fences:**
- First fence: \( y = 7x^2 \)
- Second fence: \( y = x^2 + 8 \)
2. **Finding Intersection Points:**
To find the region enclosed by these two curves, begin by finding their points of intersection. Set the equations equal to each other:
\[ 7x^2 = x^2 + 8 \]
3. **Solving the Equation:**
Subtract \( x^2 \) from both sides:
\[ 6x^2 = 8 \]
\[ x^2 = \frac{8}{6} \]
\[ x^2 = \frac{4}{3} \]
\[ x = \pm \sqrt{\frac{4}{3}} = \pm \frac{2}{\sqrt{3}} = \pm\frac{2\sqrt{3}}{3} \]
4. **Setting Up the Integral:**
The area between two curves is found using integration:
\[
A = \int_{a}^{b} \left[ f(x) - g(x) \right] \, dx
\]
Here, \( f(x) = x^2 + 8 \) and \( g(x) = 7x^2 \), with limits of integration \( a = -\frac{2\sqrt{3}}{3} \) and \( b = \frac{2\sqrt{3}}{3} \).
5. **Calculating the Area:**
\[
A = \int_{-\frac{2\sqrt{3}}{3}}^{\frac{2\sqrt{3}}{3}} \left[ (x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9659bf0-1b68-44e7-9939-e8c666150000%2F5b351135-68c0-4173-aa6a-69584387491e%2Fi5qdwr5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Mathematical Enclosures for Farm Fields**
Farmer Jones and his wife, Dr. Jones, decide to build a fence in their field to keep the sheep safe. Since Dr. Jones is a mathematician, she suggests building fences described by the equations \( y = 7x^2 \) and \( y = x^2 + 8 \). Farmer Jones thinks this would be much harder than just building an enclosure with straight sides, but he wants to please his wife.
**Problem:**
Given the equations of these curves, determine the area of the enclosed region.
**Explanation:**
1. **Equations of the Fences:**
- First fence: \( y = 7x^2 \)
- Second fence: \( y = x^2 + 8 \)
2. **Finding Intersection Points:**
To find the region enclosed by these two curves, begin by finding their points of intersection. Set the equations equal to each other:
\[ 7x^2 = x^2 + 8 \]
3. **Solving the Equation:**
Subtract \( x^2 \) from both sides:
\[ 6x^2 = 8 \]
\[ x^2 = \frac{8}{6} \]
\[ x^2 = \frac{4}{3} \]
\[ x = \pm \sqrt{\frac{4}{3}} = \pm \frac{2}{\sqrt{3}} = \pm\frac{2\sqrt{3}}{3} \]
4. **Setting Up the Integral:**
The area between two curves is found using integration:
\[
A = \int_{a}^{b} \left[ f(x) - g(x) \right] \, dx
\]
Here, \( f(x) = x^2 + 8 \) and \( g(x) = 7x^2 \), with limits of integration \( a = -\frac{2\sqrt{3}}{3} \) and \( b = \frac{2\sqrt{3}}{3} \).
5. **Calculating the Area:**
\[
A = \int_{-\frac{2\sqrt{3}}{3}}^{\frac{2\sqrt{3}}{3}} \left[ (x
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