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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![**Evaluate the Integral Using Area Formula:**
\[ \int_{2}^{3} (3 - x) \, dx. \]
**Explanation:**
This problem involves evaluating a definite integral, \(\int_{2}^{3} (3 - x) \, dx\), using the area formula.
To solve this, consider the function \(f(x) = 3 - x\), which is a linear function representing a straight line. The area under the curve from \(x = 2\) to \(x = 3\) forms a right triangle on the coordinate plane.
- **Base of the Triangle:** The length along the x-axis from 2 to 3 is 1 unit.
- **Height of the Triangle:** At \(x = 2\), the function value is \(f(2) = 3 - 2 = 1\).
- **Area of the Triangle:** The area \(A\) can be calculated using the formula for the area of a triangle:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}. \]
Therefore, the value of the definite integral is \(\frac{1}{2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb234898a-ca00-4740-ae3d-79576f672843%2Fdec7d27d-7733-4cb9-a857-df134ac835e6%2F5zs4g6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate the Integral Using Area Formula:**
\[ \int_{2}^{3} (3 - x) \, dx. \]
**Explanation:**
This problem involves evaluating a definite integral, \(\int_{2}^{3} (3 - x) \, dx\), using the area formula.
To solve this, consider the function \(f(x) = 3 - x\), which is a linear function representing a straight line. The area under the curve from \(x = 2\) to \(x = 3\) forms a right triangle on the coordinate plane.
- **Base of the Triangle:** The length along the x-axis from 2 to 3 is 1 unit.
- **Height of the Triangle:** At \(x = 2\), the function value is \(f(2) = 3 - 2 = 1\).
- **Area of the Triangle:** The area \(A\) can be calculated using the formula for the area of a triangle:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 1 \times 1 = \frac{1}{2}. \]
Therefore, the value of the definite integral is \(\frac{1}{2}\).
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