JEE Exam > JEE Questions > The area bounded by the loop of the curve 4y2...
Start Learning for Free
The area bounded by the loop of the curve 4y2 = x2 (4 – x2) is :-
Correct answer is '5.33'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-C...
Most Upvoted Answer
The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-C...
The curve 4y2=x2(4-x2) can be simplified as:
4y2=4x2 - x4
Dividing both sides by 4x2, we get:
y2 = (4 - x2)/4
Taking the square root of both sides, we get:
y = ± √((4 - x2)/4)
Since the curve is symmetric about the y-axis, we only need to consider the positive values of y. The curve intersects the x-axis at x = 0 and x = 2. So we need to find the area between the curve and the x-axis for x between 0 and 2.
To do this, we can integrate the square root expression with respect to x:
A = ∫0^2 √((4 - x2)/4) dx
We can simplify the expression under the square root by factoring out a 4:
A = ∫0^2 √(4(1 - x2/4)) dx
A = 2 ∫0^2 √(1 - x2/4) dx
We can make the substitution u = x/2, du/dx = 1/2 dx:
A = 4 ∫0^1 √(1 - u2) du
This is the area of a quarter circle with radius 1. So the final answer is:
A = 4 ∫0^1 √(1 - u2) du = 4(π/4) = π
Therefore, the area bounded by the loop of the curve 4y2=x2(4-x2) is π.
4y2=4x2 - x4
Dividing both sides by 4x2, we get:
y2 = (4 - x2)/4
Taking the square root of both sides, we get:
y = ± √((4 - x2)/4)
Since the curve is symmetric about the y-axis, we only need to consider the positive values of y. The curve intersects the x-axis at x = 0 and x = 2. So we need to find the area between the curve and the x-axis for x between 0 and 2.
To do this, we can integrate the square root expression with respect to x:
A = ∫0^2 √((4 - x2)/4) dx
We can simplify the expression under the square root by factoring out a 4:
A = ∫0^2 √(4(1 - x2/4)) dx
A = 2 ∫0^2 √(1 - x2/4) dx
We can make the substitution u = x/2, du/dx = 1/2 dx:
A = 4 ∫0^1 √(1 - u2) du
This is the area of a quarter circle with radius 1. So the final answer is:
A = 4 ∫0^1 √(1 - u2) du = 4(π/4) = π
Therefore, the area bounded by the loop of the curve 4y2=x2(4-x2) is π.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Top Courses for JEEView all
The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer?
Question Description
The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? for JEE 2024 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 loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer?.
The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? for JEE 2024 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 loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer?.
Solutions for The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer?, a detailed solution for The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? has been provided alongside types of The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The area bounded by the loop of the curve 4y2= x2(4 – x2) is :-Correct answer is '5.33'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup