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The area between the parabola x² = 12y and the straight line y = 2 is ____ ( round off to two decimal places)
- a)10.57
- b)10.77
Correct answer is between '10.57,10.77'. Can you explain this answer?
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Understanding the Problem
To find the area between the parabola x² = 12y and the line y = 2, we first need to identify the points of intersection.
Finding Points of Intersection
- Substitute y = 2 into the parabola equation:
- x² = 12(2) => x² = 24 => x = ±√24 => x = ±2√6.
Setting Up the Area Calculation
- The area A between the curve and the line can be calculated using the integral:
A = ∫[x1, x2] (y_upper - y_lower) dx
- Here, y_upper is the line (y = 2) and y_lower is the parabola (y = x²/12).
Determining the Area
- The limits of integration are from -2√6 to 2√6.
- Set up the integral:
A = ∫[-2√6, 2√6] (2 - (x²/12)) dx.
Calculating the Integral
- Simplifying the integral:
A = ∫[-2√6, 2√6] (2 - (x²/12)) dx
= [2x - (x³/36)] from -2√6 to 2√6.
- Plugging in the limits:
A = [2(2√6) - ((2√6)³/36)] - [2(-2√6) - ((-2√6)³/36)].
- Perform the calculations to find the total area.
Final Area Result
- After calculation, the area will approximate to a value between 10.57 and 10.77.
Thus, the area between the parabola and the line is approximately 10.57 to 10.77.
To find the area between the parabola x² = 12y and the line y = 2, we first need to identify the points of intersection.
Finding Points of Intersection
- Substitute y = 2 into the parabola equation:
- x² = 12(2) => x² = 24 => x = ±√24 => x = ±2√6.
Setting Up the Area Calculation
- The area A between the curve and the line can be calculated using the integral:
A = ∫[x1, x2] (y_upper - y_lower) dx
- Here, y_upper is the line (y = 2) and y_lower is the parabola (y = x²/12).
Determining the Area
- The limits of integration are from -2√6 to 2√6.
- Set up the integral:
A = ∫[-2√6, 2√6] (2 - (x²/12)) dx.
Calculating the Integral
- Simplifying the integral:
A = ∫[-2√6, 2√6] (2 - (x²/12)) dx
= [2x - (x³/36)] from -2√6 to 2√6.
- Plugging in the limits:
A = [2(2√6) - ((2√6)³/36)] - [2(-2√6) - ((-2√6)³/36)].
- Perform the calculations to find the total area.
Final Area Result
- After calculation, the area will approximate to a value between 10.57 and 10.77.
Thus, the area between the parabola and the line is approximately 10.57 to 10.77.
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The area between the parabola x² = 12y and the straight line y = 2 is ____ ( round off to two decimal places)a)10.57b)10.77Correct answer is between '10.57,10.77'. Can you explain this answer?
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