Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Question:**
Find the exact value of the integral using formulas from geometry.
\[
\int_{2}^{6} (3 + x) \, dx
\]
**Explanation:**
To solve this problem using geometry, we need to interpret the integral as the area under the function \(3 + x\) between \(x = 2\) and \(x = 6\).
1. **Graph and Shape Identification:**
- The function \(3 + x\) represents a straight line with a slope of 1 and a y-intercept of 3.
- For \(x = 2\), the function value is \(3 + 2 = 5\).
- For \(x = 6\), the function value is \(3 + 6 = 9\).
- The integral \(\int_{2}^{6} (3 + x) \, dx\) represents the area under this line from \(x = 2\) to \(x = 6\).
- The area under the line forms a trapezoid.
2. **Trapezoid Area Calculation:**
- The trapezoid has parallel sides (bases) of lengths 5 and 9 (values of the function at \(x = 2\) and \(x = 6\) respectively).
- The height (distance between \(x = 2\) and \(x = 6\)) is \(6 - 2 = 4\).
The area \(A\) of a trapezoid can be calculated using the formula:
\[
A = \frac{1}{2} \times ( \text{base}_1 + \text{base}_2 ) \times \text{height}
\]
Substituting the values we have:
\[
A = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28
\]
Thus, the exact value of the integral \(\int_{2}^{6} (3 + x) \, dx\) is 28.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe634a2f-90fc-4c70-a49c-ac4a0564f81f%2F867baf4b-516f-43bf-97bf-a2feece2e623%2Fmz4545q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Question:**
Find the exact value of the integral using formulas from geometry.
\[
\int_{2}^{6} (3 + x) \, dx
\]
**Explanation:**
To solve this problem using geometry, we need to interpret the integral as the area under the function \(3 + x\) between \(x = 2\) and \(x = 6\).
1. **Graph and Shape Identification:**
- The function \(3 + x\) represents a straight line with a slope of 1 and a y-intercept of 3.
- For \(x = 2\), the function value is \(3 + 2 = 5\).
- For \(x = 6\), the function value is \(3 + 6 = 9\).
- The integral \(\int_{2}^{6} (3 + x) \, dx\) represents the area under this line from \(x = 2\) to \(x = 6\).
- The area under the line forms a trapezoid.
2. **Trapezoid Area Calculation:**
- The trapezoid has parallel sides (bases) of lengths 5 and 9 (values of the function at \(x = 2\) and \(x = 6\) respectively).
- The height (distance between \(x = 2\) and \(x = 6\)) is \(6 - 2 = 4\).
The area \(A\) of a trapezoid can be calculated using the formula:
\[
A = \frac{1}{2} \times ( \text{base}_1 + \text{base}_2 ) \times \text{height}
\]
Substituting the values we have:
\[
A = \frac{1}{2} \times (5 + 9) \times 4 = \frac{1}{2} \times 14 \times 4 = 28
\]
Thus, the exact value of the integral \(\int_{2}^{6} (3 + x) \, dx\) is 28.
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