Find the area above curve y=(1/2*x-2)^6+5 enclosed by a line cutting at (0,y) and (x,y).
Note.It is a straight line.
Huh? It seems that you want the area under some arbitrary horizontal line.
So, let's say the line is through (0,6) and (2,6)
The area would then be (using symmetry of the region)
A = 2∫[0,2] 6 - ((1/2*x-2)^6+5) dx
That's just a simple substitution, letting
u = x/2 - 2
2du = dx
A = 2∫[-2,-1] 1-u^6 du
Answer is 438+6/7
I guess if you want an arbitrary x value, you could use this.
https://www.wolframalpha.com/input/?i=2%E2%88%AB%5B0%2Cx%5D+%28%28t%2F2-2%29%5E6+%2B+5%29+dt
Oops. That's the area under the curve, not the area above the curve. You can adjust it as needed, I guess.
Oops. I forgot to note that the axis of symmetry is x=4, not x=0. So the area would be, for some arbitrary x>=4, since the area desired is a rectangle of area 2(x-4)y, you'd get
2((x-4)*((x/2-2)^6+5)-(∫[2,x] ((t/2-2)^6 + 5) dt))
= 53/1952 (x-4)^7 - 72/7
Better double-check my math ...
To find the area above the curve y = (1/2*x-2)^6+5 enclosed by a line cutting at (0, y) and (x, y), we need to find the points of intersection between the curve and the line.
Let's start by finding the equation of the line. Since the line cuts at (0, y) and (x, y), it means that the line is a horizontal line with a y-coordinate equal to y.
Now, let's substitute the y-coordinate of the line into the equation of the curve to find the points of intersection. We have:
(1/2*x-2)^6+5 = y
Raise both sides to the power 1/6 to isolate "x":
(1/2*x-2) = (y-5)^(1/6)
Simplify:
1/2*x-2 = (y-5)^(1/6)
Now, solve for "x":
1/2*x = (y-5)^(1/6) + 2
Multiply both sides by 2 to isolate "x":
x = 2 * ((y-5)^(1/6) + 2)
These are the x-coordinates of the points of intersection between the curve and the line.
To find the area between the curve and the line, we need to integrate the curve equation minus the line equation within the interval of the x-coordinates of the points of intersection.
Integral of (curve - line) dx from point A to point B, where A is the x-coordinate of the first point of intersection and B is the x-coordinate of the second point of intersection.
The area can be calculated as:
Area = ∫[(1/2*x-2)^6+5 - y] dx from x = A to x = B
Unfortunately, without specific values for "y" and the points of intersection, we cannot calculate the exact area.