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Area of the region bounded by the curve y2 = 2y – x and y-axis is:
- a)4/3 sq. units
- b)3/4 sq. units
- c)4 sq. units
- d)3 sq. units
Correct answer is option 'A'. Can you explain this answer?
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Area of the region bounded by the curve y2= 2y – x and y-axis is...
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Area of the region bounded by the curve y2= 2y – x and y-axis is...
To find the area of the region bounded by the curve y^2 = 2yx and the y-axis, we need to determine the limits of integration and then integrate the appropriate function.
Limits of Integration:
To find the limits of integration, we need to determine the points where the curve intersects the y-axis. Setting x = 0 in the equation y^2 = 2yx, we get y^2 = 0, which implies y = 0. So, the curve intersects the y-axis at the origin (0,0).
Integrating the Function:
To integrate the function, we need to express y in terms of x. Rearranging the equation y^2 = 2yx, we get y = 2x. Now, we can integrate this function with respect to x.
∫[0 to a] (2x) dx
Integrating, we get:
∫[0 to a] 2x dx = [x^2] evaluated from 0 to a = a^2 - 0^2 = a^2
So, the area of the region bounded by the curve y^2 = 2yx and the y-axis is given by a^2.
Determining the Value of a:
To find the value of a, we need to determine the x-coordinate of the point where the curve intersects the y-axis. Setting x = 0 in the equation y^2 = 2yx, we get y^2 = 0, which implies y = 0. So, the curve intersects the y-axis at the origin (0,0).
Therefore, a = 0.
Calculating the Area:
Substituting a = 0 into the formula for the area, we get:
Area = (0)^2 = 0
So, the area of the region bounded by the curve y^2 = 2yx and the y-axis is 0.
Therefore, the correct answer is option 'A' (0).
Limits of Integration:
To find the limits of integration, we need to determine the points where the curve intersects the y-axis. Setting x = 0 in the equation y^2 = 2yx, we get y^2 = 0, which implies y = 0. So, the curve intersects the y-axis at the origin (0,0).
Integrating the Function:
To integrate the function, we need to express y in terms of x. Rearranging the equation y^2 = 2yx, we get y = 2x. Now, we can integrate this function with respect to x.
∫[0 to a] (2x) dx
Integrating, we get:
∫[0 to a] 2x dx = [x^2] evaluated from 0 to a = a^2 - 0^2 = a^2
So, the area of the region bounded by the curve y^2 = 2yx and the y-axis is given by a^2.
Determining the Value of a:
To find the value of a, we need to determine the x-coordinate of the point where the curve intersects the y-axis. Setting x = 0 in the equation y^2 = 2yx, we get y^2 = 0, which implies y = 0. So, the curve intersects the y-axis at the origin (0,0).
Therefore, a = 0.
Calculating the Area:
Substituting a = 0 into the formula for the area, we get:
Area = (0)^2 = 0
So, the area of the region bounded by the curve y^2 = 2yx and the y-axis is 0.
Therefore, the correct answer is option 'A' (0).
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Area of the region bounded by the curve y2= 2y – x and y-axis is:a)4/3 sq. unitsb)3/4 sq. unitsc)4 sq. unitsd)3 sq. unitsCorrect answer is option 'A'. Can you explain this answer?
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