Question
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.
Note.It is a straight line.
Answers
oobleck
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
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
Raj
Answer is 438+6/7
oobleck
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
https://www.wolframalpha.com/input/?i=2%E2%88%AB%5B0%2Cx%5D+%28%28t%2F2-2%29%5E6+%2B+5%29+dt
oobleck
Oops. That's the area under the curve, not the area above the curve. You can adjust it as needed, I guess.
oobleck
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 ...
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 ...