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The area bounded by the curve y = x² + 1 & the straight line x + y = 3 is:
- a)9/2
- b)4
- c)7/2
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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The area bounded by the curve y = x² + 1 & the straight line ...
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Understanding the Curves
The area bounded by the curve y = x² + 1 and the line x + y = 3 needs to be determined.
Step 1: Find the Points of Intersection
- Rearranging the line equation: y = 3 - x
- Set the equations equal: x² + 1 = 3 - x
- Rearranging gives: x² + x - 2 = 0
- Factoring: (x - 1)(x + 2) = 0
- Solutions: x = 1 and x = -2
Step 2: Determine the Corresponding y-values
- For x = 1: y = 1² + 1 = 2
- For x = -2: y = (-2)² + 1 = 5
Step 3: Sketch the Graph
- The parabola opens upwards and intersects the line at points (-2, 5) and (1, 2).
Step 4: Set Up the Integral for Area Calculation
- The area A between the curves from x = -2 to x = 1 is given by the integral:
A = ∫ from -2 to 1 [(3 - x) - (x² + 1)] dx
Step 5: Evaluate the Integral
- Simplifying the integrand: A = ∫ from -2 to 1 (2 - x - x²) dx
- Calculating the integral gives: [2x - (x²/2) - (x³/3)] from -2 to 1
Final Calculation
- Evaluating at the boundaries:
- At x = 1: 2(1) - (1/2) - (1/3) = 2 - 0.5 - 0.333 = 1.167
- At x = -2: 2(-2) - ((-2)²/2) - ((-2)³/3) = -4 - 2 + 2.667 = -3.333
- Area = 1.167 - (-3.333) = 4.5 = 9/2
Conclusion
The area bounded by the curve y = x² + 1 and the line x + y = 3 is indeed 9/2, which confirms the correct answer as option 'A'.
The area bounded by the curve y = x² + 1 and the line x + y = 3 needs to be determined.
Step 1: Find the Points of Intersection
- Rearranging the line equation: y = 3 - x
- Set the equations equal: x² + 1 = 3 - x
- Rearranging gives: x² + x - 2 = 0
- Factoring: (x - 1)(x + 2) = 0
- Solutions: x = 1 and x = -2
Step 2: Determine the Corresponding y-values
- For x = 1: y = 1² + 1 = 2
- For x = -2: y = (-2)² + 1 = 5
Step 3: Sketch the Graph
- The parabola opens upwards and intersects the line at points (-2, 5) and (1, 2).
Step 4: Set Up the Integral for Area Calculation
- The area A between the curves from x = -2 to x = 1 is given by the integral:
A = ∫ from -2 to 1 [(3 - x) - (x² + 1)] dx
Step 5: Evaluate the Integral
- Simplifying the integrand: A = ∫ from -2 to 1 (2 - x - x²) dx
- Calculating the integral gives: [2x - (x²/2) - (x³/3)] from -2 to 1
Final Calculation
- Evaluating at the boundaries:
- At x = 1: 2(1) - (1/2) - (1/3) = 2 - 0.5 - 0.333 = 1.167
- At x = -2: 2(-2) - ((-2)²/2) - ((-2)³/3) = -4 - 2 + 2.667 = -3.333
- Area = 1.167 - (-3.333) = 4.5 = 9/2
Conclusion
The area bounded by the curve y = x² + 1 and the line x + y = 3 is indeed 9/2, which confirms the correct answer as option 'A'.
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