Find the centroid of the region bounded by the graphs of the functions y=72².y= 2² +3 The centroid is at (7,5) where
Find the centroid of the region bounded by the graphs of the functions y=72².y= 2² +3 The centroid is at (7,5) where
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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![**Finding the Centroid of a Region Bounded by Two Functions**
When determining the centroid (or geometric center) of a region bounded by two curves, we typically refer to the area in the coordinate plane enclosed by the graphs of the functions. For the given functions \( y = 7x^2 \) and \( y = x^2 + 3 \), we aim to find the centroid of the region they enclose.
**Problem Statement**
Find the centroid of the region bounded by the graphs of the functions \( y = 7x^2 \) and \( y = x^2 + 3 \).
**Centroid Formula**
The centroid \((\bar{x}, \bar{y})\) for a region bounded by two curves can be computed using the following formulas:
\[ \bar{x} = \frac{1}{A} \int_{a}^{b} x [f(x) - g(x)] \, dx \]
\[ \bar{y} = \frac{1}{2A} \int_{a}^{b} [f(x) + g(x)][f(x) - g(x)] \, dx \]
where \( f(x) \) and \( g(x) \) are the upper and lower functions respectively, and \( A \) is the area of the bounded region given by:
\[ A = \int_{a}^{b} [f(x) - g(x)] \, dx \]
Once the integration limits \( a \) and \( b \) are identified (which are the x-values where the functions intersect), these integrals can be evaluated.
**Graph Analysis**
The region bounded by the graphs \( y = 7x^2 \) and \( y = x^2 + 3 \) is determined by finding the points of intersection and the area between the curves.
**Calculating Centroid Coordinates**
1. **Determine Points of Intersection**:
Solve for \( x \) where \( 7x^2 = x^2 + 3 \).
2. **Compute the Area \( A \)**:
Integrate the difference between the functions over the interval \( [a, b] \).
3. **Compute \( \bar{x} \)**:
Use the first centroid formula to find the x-coordinate of the centroid.
4. **Compute \( \bar{y} \)**:
Use the second centroid formula to find the y-coordinate](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2386874c-e724-4957-85a1-36ae7eec2eac%2F904eb359-a16d-419b-8b51-57d33b23232d%2Fdbqd5i6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the Centroid of a Region Bounded by Two Functions**
When determining the centroid (or geometric center) of a region bounded by two curves, we typically refer to the area in the coordinate plane enclosed by the graphs of the functions. For the given functions \( y = 7x^2 \) and \( y = x^2 + 3 \), we aim to find the centroid of the region they enclose.
**Problem Statement**
Find the centroid of the region bounded by the graphs of the functions \( y = 7x^2 \) and \( y = x^2 + 3 \).
**Centroid Formula**
The centroid \((\bar{x}, \bar{y})\) for a region bounded by two curves can be computed using the following formulas:
\[ \bar{x} = \frac{1}{A} \int_{a}^{b} x [f(x) - g(x)] \, dx \]
\[ \bar{y} = \frac{1}{2A} \int_{a}^{b} [f(x) + g(x)][f(x) - g(x)] \, dx \]
where \( f(x) \) and \( g(x) \) are the upper and lower functions respectively, and \( A \) is the area of the bounded region given by:
\[ A = \int_{a}^{b} [f(x) - g(x)] \, dx \]
Once the integration limits \( a \) and \( b \) are identified (which are the x-values where the functions intersect), these integrals can be evaluated.
**Graph Analysis**
The region bounded by the graphs \( y = 7x^2 \) and \( y = x^2 + 3 \) is determined by finding the points of intersection and the area between the curves.
**Calculating Centroid Coordinates**
1. **Determine Points of Intersection**:
Solve for \( x \) where \( 7x^2 = x^2 + 3 \).
2. **Compute the Area \( A \)**:
Integrate the difference between the functions over the interval \( [a, b] \).
3. **Compute \( \bar{x} \)**:
Use the first centroid formula to find the x-coordinate of the centroid.
4. **Compute \( \bar{y} \)**:
Use the second centroid formula to find the y-coordinate
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