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The area bounded by the curve y = 4x - x2 and x-axis is
- a)30/7 sq. units
- b)31/7 sq. units
- c)32/3 sq. units
- d)34/3 sq. units
Correct answer is option 'C'. Can you explain this answer?
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The area bounded by the curve y = 4x - x2 and x-axis isa)30/7 sq. unit...
To find the area bounded by the curve y = 4x - x^2 and the x-axis, we need to integrate the function between the appropriate limits.
Step 1: Find the x-coordinates of the points where the curve intersects the x-axis.
To find the points of intersection, we set y = 0 and solve for x:
0 = 4x - x^2
x^2 - 4x = 0
x(x - 4) = 0
x = 0 or x = 4
So, the curve intersects the x-axis at x = 0 and x = 4.
Step 2: Determine the limits of integration.
Since the curve intersects the x-axis at x = 0 and x = 4, these will be the limits of integration.
Step 3: Set up the integral.
The area bounded by the curve and the x-axis can be calculated using the definite integral:
A = ∫[a,b] f(x) dx
In this case, f(x) = 4x - x^2 and the limits of integration are a = 0 and b = 4. So, the integral becomes:
A = ∫[0,4] (4x - x^2) dx
Step 4: Evaluate the integral.
To evaluate the integral, we need to find the antiderivative of the function 4x - x^2. The antiderivative is:
F(x) = 2x^2 - (1/3)x^3
Using the Fundamental Theorem of Calculus, the definite integral becomes:
A = F(b) - F(a)
A = [2(4)^2 - (1/3)(4)^3] - [2(0)^2 - (1/3)(0)^3]
A = [32 - (64/3)] - [0]
A = 32/3
Therefore, the area bounded by the curve y = 4x - x^2 and the x-axis is 32/3 square units. Hence, the correct answer is option C.
Step 1: Find the x-coordinates of the points where the curve intersects the x-axis.
To find the points of intersection, we set y = 0 and solve for x:
0 = 4x - x^2
x^2 - 4x = 0
x(x - 4) = 0
x = 0 or x = 4
So, the curve intersects the x-axis at x = 0 and x = 4.
Step 2: Determine the limits of integration.
Since the curve intersects the x-axis at x = 0 and x = 4, these will be the limits of integration.
Step 3: Set up the integral.
The area bounded by the curve and the x-axis can be calculated using the definite integral:
A = ∫[a,b] f(x) dx
In this case, f(x) = 4x - x^2 and the limits of integration are a = 0 and b = 4. So, the integral becomes:
A = ∫[0,4] (4x - x^2) dx
Step 4: Evaluate the integral.
To evaluate the integral, we need to find the antiderivative of the function 4x - x^2. The antiderivative is:
F(x) = 2x^2 - (1/3)x^3
Using the Fundamental Theorem of Calculus, the definite integral becomes:
A = F(b) - F(a)
A = [2(4)^2 - (1/3)(4)^3] - [2(0)^2 - (1/3)(0)^3]
A = [32 - (64/3)] - [0]
A = 32/3
Therefore, the area bounded by the curve y = 4x - x^2 and the x-axis is 32/3 square units. Hence, the correct answer is option C.
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The area bounded by the curve y = 4x - x2 and x-axis isa)30/7 sq. unitsb)31/7 sq. unitsc)32/3 sq. unitsd)34/3 sq. unitsCorrect answer is option 'C'. Can you explain this answer?
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