Fiind the area bounded by the parabola y^2 = 8x, the x axis and the line x=2
I think the answer is 2.25V.
a = ∫[0,2] √(8x) dx
= √8 (2/3 x^3/2) [0,2]
= 2√8/3 (2√2)
= 16/3
a = ∫[0,4] 2-y^2/8 dy
= 4y - y^3/24 [0,2]
= 8 - 64/24
= 16/3
To find the area bounded by the parabola y^2 = 8x, the x-axis, and the line x = 2, we need to integrate the function that represents the area between these curves.
Step 1: Determine the intersection points
To find the intersection points between the parabola and the line x = 2, substitute x = 2 into the equation of the parabola:
y^2 = 8(2)
y^2 = 16
y = ±√16
y = ±4
Step 2: Define the integral bounds
Since we only want to find the area in the positive y-axis (above the x-axis), we will consider y = 4 as the upper limit, and y = 0 as the lower limit.
Step 3: Set up the integral
Since we are in terms of y, we need to rewrite the equation for x in terms of y:
y^2 = 8x
x = y^2/8
The area integral is given by:
Area = ∫[0 to 4] (x) dy
Step 4: Evaluate the integral
Integrating the function x with respect to y:
Area = ∫[0 to 4] (y^2/8) dy
= (1/8) ∫[0 to 4] (y^2) dy
= (1/8) * [y^3/3] [0 to 4]
= (1/8) * [(4^3/3) - (0^3/3)]
= (1/8) * [64/3]
= 8/3
Step 5: Finalize the answer
The area bounded by the parabola y^2 = 8x, the x-axis, and the line x = 2 is 8/3 square units.
Therefore, the area is 8/3 square units.
To find the area bounded by the parabola y^2 = 8x, the x-axis, and the line x = 2, we need to integrate the equation of the parabola.
Step 1: Solve the equation y^2 = 8x for y.
- Take the square root of both sides of the equation: y = ±√(8x).
- Since we are only concerned with the area between the parabola and the x-axis, we consider the positive square root: y = √(8x).
Step 2: Set up the integral.
- To find the area between the x-axis and the parabola, we need to integrate the y-coordinate expression with respect to x. The limits of integration will be the x-values that define the region.
- Since the region is bounded by the parabola y^2 = 8x, the x-axis (y = 0), and the line x = 2, we can set up the integral as follows:
∫[from x = 0 to x = 2] √(8x) dx.
Step 3: Solve the integral.
- Integrate the expression: ∫√(8x) dx.
- Use the power rule to integrate the expression: ∫x^(1/2) dx = (2/3)x^(3/2).
- Evaluate the integral at the upper and lower limits: (2/3)x^(3/2) evaluated from x = 0 to x = 2.
- Plug in the values: [(2/3)(2)^(3/2)] - [(2/3)(0)^(3/2)].
- Simplify: [(2/3)(2√2)] - 0.
- The final result is: (4/3)√2.
Therefore, the area bounded by the parabola y^2 = 8x, the x-axis, and the line x = 2 is (4/3)√2 square units.