Find the area bounded by the parabola 8+2y-y^2 and the lines y=-1 y=3
I just noticed that you did not specify that the y-axis formed part of the boundary. Without that condition, the horizontal strips extend leftward forever.
Of course, you may have mixed up x and y in places, so ...
To find the area bounded by the parabola and the lines, we need to find the points where the parabola intersects with the lines.
First, let's find the points of intersection between the parabola and the line y = -1:
Set y equal to -1 in the equation of the parabola:
8 + 2(-1) - (-1)^2 = 8 - 2 + 1 = 7
So, the parabola intersects the line y = -1 at the point (7, -1).
Now, let's find the points of intersection between the parabola and the line y = 3:
Set y equal to 3 in the equation of the parabola:
8 + 2(3) - 3^2 = 8 + 6 - 9 = 5
So, the parabola intersects the line y = 3 at the point (5, 3).
To find the area bounded by the parabola and the lines, we need to find the integral of the function that represents the parabola over the interval between the x-coordinates of the points of intersection.
The equation of the parabola is 8 + 2y - y^2. To find the x-coordinates of the points of intersection, we need to solve the equation of the parabola for x.
Rearranging the equation, we get: y^2 - 2y + 8 = 0
Solving this quadratic equation, we find two solutions:
y = (2 ± √(2^2 - 4(1)(8))) / 2
y = (2 ± √(4 - 32)) / 2
y = (2 ± √(-28)) / 2
Since we are looking for real solutions, we discard the negative square root value.
y = (2 + √(-28)) / 2
The expression inside the square root, -28, is negative which means there are no real solutions.
Therefore, there are no other points of intersection between the parabola and the line y = -1 or y = 3.
Now that we know the x-coordinates of the points of intersection (5 and 7), we can find the area bounded by the parabola and the lines by integrating the function with respect to x over the interval [5, 7].
The area can be calculated as:
Area = ∫[5,7] (8 + 2y - y^2) dx
Simplify the equation: Area = ∫[5,7] (8 + 2y - y^2) dx
Evaluate the integral: Area = [8x + 2yx - (1/3)y^3] from 5 to 7
Substitute y with the equation of each of the lines: y = -1 and y = 3
For y = -1, the area calculation becomes: Area = [8x + 2x(-1) - (1/3)(-1)^3] from 5 to 7
Simplify: Area = [8x - 2 - (1/3)] from 5 to 7
For y = 3, the area calculation becomes: Area = [8x + 2x(3) - (1/3)(3)^3] from 5 to 7
Simplify: Area = [8x + 6 - (1/3)(27)] from 5 to 7
Now, substitute the values of x into the area equations and subtract the result of the lower limit from the upper limit:
Area = [8(7) - 2 - (1/3)] - [8(5) - 2 - (1/3)]
Simplify the equation to find the area bounded by the parabola and the lines.
Note: If you evaluate the integral and find the area, make sure to check if the area is positive or negative, as it will indicate whether the area is above or below the x-axis.
This is most easily done by integrating on y, using horizontal strips.
a = ∫[-1,3] x dy
= ∫[-1,3] (8+2y-y^2) dy