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The area bounded by the parabola y = x2 and the line y = x is
- a)1/2 sq. units
- b)1/3 sq. units
- c)1/6 sq. units
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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The area bounded by the parabola y =x2and the line y = x isa)1/2 sq. u...
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The area bounded by the parabola y =x2and the line y = x isa)1/2 sq. u...
Understanding the Area Bounded by the Curves
To find the area bounded by the parabola y = x² and the line y = x, we first need to determine their points of intersection.
Step 1: Finding Points of Intersection
- Set the equations equal to each other:
x² = x
- Rearranging gives:
x² - x = 0
- Factoring the expression yields:
x(x - 1) = 0
- Thus, the points of intersection are:
x = 0 and x = 1.
Step 2: Setting Up the Area Integral
- The area between the curves from x = 0 to x = 1 can be calculated using the integral:
Area = ∫ (top curve - bottom curve) dx
- Here, the top curve is y = x, and the bottom curve is y = x².
Step 3: Calculating the Area
- The integral becomes:
Area = ∫ from 0 to 1 (x - x²) dx
- Breaking it down:
- The integral of x is (1/2)x²
- The integral of x² is (1/3)x³
- Evaluating the integral:
Area = [(1/2)x² - (1/3)x³] from 0 to 1
- At x = 1:
(1/2)(1)² - (1/3)(1)³ = 1/2 - 1/3
- Finding a common denominator (6):
(3/6) - (2/6) = 1/6
Conclusion
The area bounded by the parabola y = x² and the line y = x is 1/6 square units. Thus, the correct answer is option 'C'.
To find the area bounded by the parabola y = x² and the line y = x, we first need to determine their points of intersection.
Step 1: Finding Points of Intersection
- Set the equations equal to each other:
x² = x
- Rearranging gives:
x² - x = 0
- Factoring the expression yields:
x(x - 1) = 0
- Thus, the points of intersection are:
x = 0 and x = 1.
Step 2: Setting Up the Area Integral
- The area between the curves from x = 0 to x = 1 can be calculated using the integral:
Area = ∫ (top curve - bottom curve) dx
- Here, the top curve is y = x, and the bottom curve is y = x².
Step 3: Calculating the Area
- The integral becomes:
Area = ∫ from 0 to 1 (x - x²) dx
- Breaking it down:
- The integral of x is (1/2)x²
- The integral of x² is (1/3)x³
- Evaluating the integral:
Area = [(1/2)x² - (1/3)x³] from 0 to 1
- At x = 1:
(1/2)(1)² - (1/3)(1)³ = 1/2 - 1/3
- Finding a common denominator (6):
(3/6) - (2/6) = 1/6
Conclusion
The area bounded by the parabola y = x² and the line y = x is 1/6 square units. Thus, the correct answer is option 'C'.
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The area bounded by the parabola y =x2and the line y = x isa)1/2 sq. unitsb)1/3 sq. unitsc)1/6 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The area bounded by the parabola y =x2and the line y = x isa)1/2 sq. unitsb)1/3 sq. unitsc)1/6 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area bounded by the parabola y =x2and the line y = x isa)1/2 sq. unitsb)1/3 sq. unitsc)1/6 sq. unitsd)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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