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Area lying in the first quadrant and bounded by the circle x2 + y2 = 4, the line x = y√3 and x−axis is
- a)π
- b)π/2
- c)π/3
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Area lying in the first quadrant and bounded by the circlex2 + y2 = 4,...
Circle: x2 + y2 = 4
Line: x = y√3
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Area lying in the first quadrant and bounded by the circlex2 + y2 = 4,...
To find the area bounded by the circle x^2 + y^2 = 4 and the line x = y in the first quadrant, we need to find the points of intersection between the circle and the line.
First, let's solve the system of equations:
x = y
x^2 + y^2 = 4
Substituting x = y into the second equation:
(y)^2 + y^2 = 4
2y^2 = 4
y^2 = 2
y = √2 or y = -√2
Since we are only interested in the first quadrant, we take y = √2.
Now, we can find the x-coordinate of the intersection point by substituting y = √2 into x = y:
x = √2
So, the points of intersection are (√2, √2) and (-√2, -√2).
To find the area bounded by the circle and the line in the first quadrant, we can integrate the equation of the circle with respect to y from 0 to √2 and subtract the area of the triangle formed by the points of intersection.
The equation of the circle in terms of y is:
x = √(4 - y^2)
The area of the circle bounded by the line x = y is given by the integral:
A = ∫[0,√2] (√(4 - y^2)) dy
Using the formula for the area of a circle sector, we can rewrite the integral as:
A = 1/2 ∫[0,√2] (√(4 - y^2))^2 dy
Simplifying:
A = 1/2 ∫[0,√2] (4 - y^2) dy
A = 1/2 [4y - (y^3)/3] evaluated from 0 to √2
A = 1/2 [(4√2 - (√2)^3/3) - (0 - 0)]
A = 1/2 [4√2 - 2√2/3]
A = 1/2 [8√2/3]
A = 4√2/3
Therefore, the area bounded by the circle x^2 + y^2 = 4 and the line x = y in the first quadrant is 4√2/3.
First, let's solve the system of equations:
x = y
x^2 + y^2 = 4
Substituting x = y into the second equation:
(y)^2 + y^2 = 4
2y^2 = 4
y^2 = 2
y = √2 or y = -√2
Since we are only interested in the first quadrant, we take y = √2.
Now, we can find the x-coordinate of the intersection point by substituting y = √2 into x = y:
x = √2
So, the points of intersection are (√2, √2) and (-√2, -√2).
To find the area bounded by the circle and the line in the first quadrant, we can integrate the equation of the circle with respect to y from 0 to √2 and subtract the area of the triangle formed by the points of intersection.
The equation of the circle in terms of y is:
x = √(4 - y^2)
The area of the circle bounded by the line x = y is given by the integral:
A = ∫[0,√2] (√(4 - y^2)) dy
Using the formula for the area of a circle sector, we can rewrite the integral as:
A = 1/2 ∫[0,√2] (√(4 - y^2))^2 dy
Simplifying:
A = 1/2 ∫[0,√2] (4 - y^2) dy
A = 1/2 [4y - (y^3)/3] evaluated from 0 to √2
A = 1/2 [(4√2 - (√2)^3/3) - (0 - 0)]
A = 1/2 [4√2 - 2√2/3]
A = 1/2 [8√2/3]
A = 4√2/3
Therefore, the area bounded by the circle x^2 + y^2 = 4 and the line x = y in the first quadrant is 4√2/3.
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Area lying in the first quadrant and bounded by the circlex2 + y2 = 4,the linex = y√3 andx−axis isa)πb)π/2c)π/3d)none of theseCorrect answer is option 'C'. Can you explain this answer?
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