Question:
The value of \(\int_{-1}^{1}|x - 3|dx\) is equal to
The value of \(\int_{-1}^{1}|x - 3|dx\) is equal to
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When the integrand contains an absolute value, split the integral at the point where the expression inside changes sign.
Updated On: Apr 24, 2026
- 5
- 6
- -5
- -6
- 0
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The Correct Option is B
Solution and Explanation
Step 1: Concept:
• For \(x \in [-1, 1]\), we have: \[ x - 3<0 \]
• So, \[ |x - 3| = -(x - 3) = 3 - x \]
Step 2: Detailed Explanation:
• Evaluate the integral: \[ \int_{-1}^{1} |x - 3| \, dx = \int_{-1}^{1} (3 - x) \, dx \]
• Integrate: \[ = \left[3x - \frac{x^2}{2}\right]_{-1}^{1} \]
• Substitute limits: \[ = \left(3(1) - \frac{1}{2}\right) - \left(3(-1) - \frac{1}{2}\right) \]
• Simplify: \[ = (3 - 0.5) - (-3 - 0.5) = 2.5 - (-3.5) = 2.5 + 3.5 = 6 \]
Step 3: Final Answer:
• The value is \(6\).
• For \(x \in [-1, 1]\), we have: \[ x - 3<0 \]
• So, \[ |x - 3| = -(x - 3) = 3 - x \]
Step 2: Detailed Explanation:
• Evaluate the integral: \[ \int_{-1}^{1} |x - 3| \, dx = \int_{-1}^{1} (3 - x) \, dx \]
• Integrate: \[ = \left[3x - \frac{x^2}{2}\right]_{-1}^{1} \]
• Substitute limits: \[ = \left(3(1) - \frac{1}{2}\right) - \left(3(-1) - \frac{1}{2}\right) \]
• Simplify: \[ = (3 - 0.5) - (-3 - 0.5) = 2.5 - (-3.5) = 2.5 + 3.5 = 6 \]
Step 3: Final Answer:
• The value is \(6\).
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