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The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.?
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The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, ...
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The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, ...
Understanding the Curves and Lines
To find the area bounded by the curves y = √x and the line 2y - x - 3 = 0, we first rewrite the line equation for clarity:
- Rearranging gives: y = (x + 3)/2.
Identifying Points of Intersection
Next, we find the points where these two curves intersect in the first quadrant:
1. Set √x = (x + 3)/2.
2. Squaring both sides results in: x = (x + 3)²/4.
3. Expanding and solving leads to the quadratic equation: x² - 6x + 9 = 0.
4. This simplifies to (x - 3)² = 0, giving x = 3.
Now, substituting x = 3 back into either equation gives us y = √3.
Setting Up the Area Integral
The area A bounded by the curves can now be calculated by integrating from x = 0 to x = 3:
- A = ∫[0 to 3] (upper curve - lower curve) dx.
- Here, the upper curve is y = (x + 3)/2 and the lower curve is y = √x.
Calculating the Area
1. The integral becomes:
A = ∫[0 to 3] ((x + 3)/2 - √x) dx.
2. Solving this integral step by step:
- The integral of (x + 3)/2 is (1/4)x² + (3/2)x.
- The integral of √x is (2/3)x^(3/2).
3. Evaluating from 0 to 3 gives:
A = [(1/4)(3)² + (3/2)(3)] - [(2/3)(3)^(3/2)].
After calculating, the area simplifies to 9 square units.
Conclusion
Thus, the area bounded by the curves in the first quadrant is indeed 9 square units, confirming option c.) 9.
To find the area bounded by the curves y = √x and the line 2y - x - 3 = 0, we first rewrite the line equation for clarity:
- Rearranging gives: y = (x + 3)/2.
Identifying Points of Intersection
Next, we find the points where these two curves intersect in the first quadrant:
1. Set √x = (x + 3)/2.
2. Squaring both sides results in: x = (x + 3)²/4.
3. Expanding and solving leads to the quadratic equation: x² - 6x + 9 = 0.
4. This simplifies to (x - 3)² = 0, giving x = 3.
Now, substituting x = 3 back into either equation gives us y = √3.
Setting Up the Area Integral
The area A bounded by the curves can now be calculated by integrating from x = 0 to x = 3:
- A = ∫[0 to 3] (upper curve - lower curve) dx.
- Here, the upper curve is y = (x + 3)/2 and the lower curve is y = √x.
Calculating the Area
1. The integral becomes:
A = ∫[0 to 3] ((x + 3)/2 - √x) dx.
2. Solving this integral step by step:
- The integral of (x + 3)/2 is (1/4)x² + (3/2)x.
- The integral of √x is (2/3)x^(3/2).
3. Evaluating from 0 to 3 gives:
A = [(1/4)(3)² + (3/2)(3)] - [(2/3)(3)^(3/2)].
After calculating, the area simplifies to 9 square units.
Conclusion
Thus, the area bounded by the curves in the first quadrant is indeed 9 square units, confirming option c.) 9.
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The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.?.
The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area(square units) bounded by the curves y=√x, and line 2y-x 3=0, x-axis and lying in first quadrant is: a.) 18 b.) 27/4 c.) 9 d.) 36 Correct answer is c.) 9. Explain.?.
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