The region bounded by the given curves is rotated about the specified axis. find the volume v of the resulting solid by any method. y = −x2 + 14x − 45, y = 0; about the x-axis

Step-by-step explanation:

First thing I did was graph this on my calculator to find the upper and lower bounds.  They were x = 5, 9.  Then I applied the disk method of integration.  From this I know know I use y = x equations, x intervals, I need a value for R, and then I need a value for r.  R represents the height of the representative rectangle which is perpendicular to the axis of rotation, so the height of that rectangle starts at y = 0 and meets the curve.  So

Since there is no "hole" or space between the graph and the axis of rotation, r=0.  Putting that into the volume formula:

To simplify I have to multiply that polynomial by itself.  Doing that gives us:

Integrating gives us:

which we integrate from 5 to 9

Doing a bit of simplification on that:

from 5 to 9

Applying the First Fundamental Theorem of Calculus we get that when we sub in a 9 for x then a 5 for x:

which subtracts to