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The area bounded by the x-axis and the curve y = 4x - x2 - 3 is
- a)4/3
- b)3/4
- c)7
- d)3/2
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
The area bounded by the x-axis and the curve y = 4x - x2 - 3 isa)4/3b)...
Given, the equation of the curve is y = 4x - x² - 3.
To find the area bounded by the x-axis and the curve, we need to integrate the curve with respect to x from the points where the curve intersects the x-axis.
Step-by-Step Solution:
1. Finding the x-intercepts:
When y = 0, we have:
0 = 4x - x² - 3
x² - 4x + 3 = 0
(x - 1)(x - 3) = 0
Therefore, the curve intersects the x-axis at x = 1 and x = 3.
2. Integrating the curve:
We need to integrate the curve with respect to x from x = 1 to x = 3 to find the area bounded by the curve and the x-axis. So, we have:
Area = ∫₁³ [4x - x² - 3] dx
= [2x² - (1/3)x³ - 3x]₁³
= [2(3)² - (1/3)(3)³ - 3(3)] - [2(1)² - (1/3)(1)³ - 3(1)]
= (18 - 9 - 9) - (2 - (1/3) - 3)
= (-3/3) - (7/3)
= -10/3
However, we are interested in the area, which is always positive. So, we take the absolute value of the result.
Therefore, the area bounded by the curve and the x-axis is:
| (-10/3) | = 10/3 = 3.33 (approx)
Hence, the correct option is (a) 4/3.
To find the area bounded by the x-axis and the curve, we need to integrate the curve with respect to x from the points where the curve intersects the x-axis.
Step-by-Step Solution:
1. Finding the x-intercepts:
When y = 0, we have:
0 = 4x - x² - 3
x² - 4x + 3 = 0
(x - 1)(x - 3) = 0
Therefore, the curve intersects the x-axis at x = 1 and x = 3.
2. Integrating the curve:
We need to integrate the curve with respect to x from x = 1 to x = 3 to find the area bounded by the curve and the x-axis. So, we have:
Area = ∫₁³ [4x - x² - 3] dx
= [2x² - (1/3)x³ - 3x]₁³
= [2(3)² - (1/3)(3)³ - 3(3)] - [2(1)² - (1/3)(1)³ - 3(1)]
= (18 - 9 - 9) - (2 - (1/3) - 3)
= (-3/3) - (7/3)
= -10/3
However, we are interested in the area, which is always positive. So, we take the absolute value of the result.
Therefore, the area bounded by the curve and the x-axis is:
| (-10/3) | = 10/3 = 3.33 (approx)
Hence, the correct option is (a) 4/3.
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Community Answer
The area bounded by the x-axis and the curve y = 4x - x2 - 3 isa)4/3b)...
You can solve it by first finding the points where the curve is cutting the x-axis then integrate it from the point to the other.
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