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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calculate this equation
![The image appears to show a handwritten mathematical integral on graph paper. Here is the transcription of the content:
---
### Calculate
\[ \int_{0}^{2} 14(x+1)^2 \, dx \]
---
#### Explanation:
The given integral is a definite integral from 0 to 2 of the function \( 14(x+1)^2 \) with respect to \( x \). To solve this integral:
1. **Expand the Integrand**: First, expand \( (x+1)^2 \).
\[ (x+1)^2 = x^2 + 2x + 1 \]
2. **Distribute the Constant**: Multiply the expansion by 14.
\[ 14(x^2 + 2x + 1) = 14x^2 + 28x + 14 \]
3. **Integrate the Expanded Function**:
\[ \int_{0}^{2} (14x^2 + 28x + 14) \, dx \]
4. **Apply the Power Rule of Integration**: Integrate each term separately.
\[ \int_{0}^{2} 14x^2 \, dx = 14 \left[\frac{x^3}{3}\right]_{0}^{2} \]
\[ \int_{0}^{2} 28x \, dx = 28 \left[\frac{x^2}{2}\right]_{0}^{2} \]
\[ \int_{0}^{2} 14 \, dx = 14 [x]_{0}^{2} \]
5. **Evaluate the Integrals**:
\[ 14 \left[\frac{x^3}{3}\right]_{0}^{2} = 14 \left(\frac{2^3}{3} - \frac{0^3}{3}\right) = 14 \left(\frac{8}{3} - 0\right) = \frac{112}{3} \]
\[ 28 \left[\frac{x^2}{2}\right]_{0}^{2} = 28 \left(\frac{2^2}{2} - \frac{0^2}{2}\right) = 28 \left(2 - 0\right) = 56 \]
\[ 14 [x]_{0}^{2} = 14 (2 - 0) = 28 \]
6.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5886f279-1ae3-47c9-a907-c1e5911c0685%2F685d5da7-a05f-4a97-b22e-72ebb0a39a89%2F07rxvr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image appears to show a handwritten mathematical integral on graph paper. Here is the transcription of the content:
---
### Calculate
\[ \int_{0}^{2} 14(x+1)^2 \, dx \]
---
#### Explanation:
The given integral is a definite integral from 0 to 2 of the function \( 14(x+1)^2 \) with respect to \( x \). To solve this integral:
1. **Expand the Integrand**: First, expand \( (x+1)^2 \).
\[ (x+1)^2 = x^2 + 2x + 1 \]
2. **Distribute the Constant**: Multiply the expansion by 14.
\[ 14(x^2 + 2x + 1) = 14x^2 + 28x + 14 \]
3. **Integrate the Expanded Function**:
\[ \int_{0}^{2} (14x^2 + 28x + 14) \, dx \]
4. **Apply the Power Rule of Integration**: Integrate each term separately.
\[ \int_{0}^{2} 14x^2 \, dx = 14 \left[\frac{x^3}{3}\right]_{0}^{2} \]
\[ \int_{0}^{2} 28x \, dx = 28 \left[\frac{x^2}{2}\right]_{0}^{2} \]
\[ \int_{0}^{2} 14 \, dx = 14 [x]_{0}^{2} \]
5. **Evaluate the Integrals**:
\[ 14 \left[\frac{x^3}{3}\right]_{0}^{2} = 14 \left(\frac{2^3}{3} - \frac{0^3}{3}\right) = 14 \left(\frac{8}{3} - 0\right) = \frac{112}{3} \]
\[ 28 \left[\frac{x^2}{2}\right]_{0}^{2} = 28 \left(\frac{2^2}{2} - \frac{0^2}{2}\right) = 28 \left(2 - 0\right) = 56 \]
\[ 14 [x]_{0}^{2} = 14 (2 - 0) = 28 \]
6.
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