College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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![**Finding the X Component of the Centroid**
In this exercise, we will determine the x-component of the centroid for a given region. The region is defined by the curve \( y = x^{2/3} \) and is bounded by the x-axis from \( x = 0 \) to \( x = 8 \) inches.
**Diagram Explanation:**
The diagram shows an area under the curve \( y = x^{2/3} \), between \( x = 0 \) and \( x = 8 \) inches. Additionally:
- The y-axis extends up to 4 inches.
- The x-axis extends up to 8 inches.
- The shaded area represents the region for which we will find the centroid.
To find the x-component of the centroid \( \overline{x} \), we will use the following formula:
\[ \overline{x} = \frac{\int_a^b x \, f(x) \, dx}{\int_a^b f(x) \, dx} \]
Here:
- The limits of integration are from \( a = 0 \) to \( b = 8 \).
- The function \( f(x) = x^{2/3} \).
Next, we evaluate the integrals:
1. **Numerator:**
\[ \int_0^8 x \, (x^{2/3}) \, dx = \int_0^8 x^{5/3} \, dx \]
\[ = \left[ \frac{3}{8} x^{8/3} \right]_0^8 \]
\[ = \frac{3}{8} \left(8^{8/3} - 0 \right) \]
\[ = \frac{3}{8} (512) \]
\[ = 192 \]
2. **Denominator:**
\[ \int_0^8 x^{2/3} \, dx = \left[ \frac{3}{5} x^{5/3} \right]_0^8 \]
\[ = \frac{3}{5} \left(8^{5/3} - 0 \right) \]
\[ = \frac{3}{5} (32) \]
\[ = 19.2 \]
Thus, the x-component of the centroid is:
\[ \overline{x} = \frac{192}{19.2}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe3513363-cb6f-4339-8b85-3b7b98137f18%2F83ec4b86-f65d-4d32-b918-2fc6d9798ccb%2Fxmfckx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding the X Component of the Centroid**
In this exercise, we will determine the x-component of the centroid for a given region. The region is defined by the curve \( y = x^{2/3} \) and is bounded by the x-axis from \( x = 0 \) to \( x = 8 \) inches.
**Diagram Explanation:**
The diagram shows an area under the curve \( y = x^{2/3} \), between \( x = 0 \) and \( x = 8 \) inches. Additionally:
- The y-axis extends up to 4 inches.
- The x-axis extends up to 8 inches.
- The shaded area represents the region for which we will find the centroid.
To find the x-component of the centroid \( \overline{x} \), we will use the following formula:
\[ \overline{x} = \frac{\int_a^b x \, f(x) \, dx}{\int_a^b f(x) \, dx} \]
Here:
- The limits of integration are from \( a = 0 \) to \( b = 8 \).
- The function \( f(x) = x^{2/3} \).
Next, we evaluate the integrals:
1. **Numerator:**
\[ \int_0^8 x \, (x^{2/3}) \, dx = \int_0^8 x^{5/3} \, dx \]
\[ = \left[ \frac{3}{8} x^{8/3} \right]_0^8 \]
\[ = \frac{3}{8} \left(8^{8/3} - 0 \right) \]
\[ = \frac{3}{8} (512) \]
\[ = 192 \]
2. **Denominator:**
\[ \int_0^8 x^{2/3} \, dx = \left[ \frac{3}{5} x^{5/3} \right]_0^8 \]
\[ = \frac{3}{5} \left(8^{5/3} - 0 \right) \]
\[ = \frac{3}{5} (32) \]
\[ = 19.2 \]
Thus, the x-component of the centroid is:
\[ \overline{x} = \frac{192}{19.2}
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