# Rotating on a slant

1. Dec 6, 2009

### Samuelb88

1. The problem statement, all variables and given/known data
Let C be the arc of the curve y = f(x) between points P(p,f(p)), Q(q,f(q)), and let R be the region bounded by C, the line y = mx + b, and the perpendiculars by the line from P and Q.

http://img15.imageshack.us/img15/6827/pic1ds.jpg [Broken]

Show that the area R is

$$\frac{1}{1+m^2}\right) \int_q^p (f(x) - mx - b)(1+m\frac{df}{dx}\right) ) dx$$

Hint: This formula can be verified by subtracting areas, but it will be helpful to derive region R using rectangles perpendicular to the line, shown in the figure below.

http://img163.imageshack.us/img163/1365/pic2c.jpg [Broken]

2. Relevant equations

3. The attempt at a solution
Tangent to C @ (x_i,f(x_i)): $$y=f'(x_i)(x-x_i) + f(x_i)$$

I let L equal the line segment "?" perpendicular to the line y=mx+b and used the distance equation to determine its value.

$$L = ((x_i-x)^2+(f'(x_i)(x-x_i) + f(x_i) - mx - b)^2)^(^1^/^2^)$$

And the area A

A = L(change in u)

So the region R should equal the limit of all the approximating rectangles.

Pretty confused as of how to proceed. Any suggestions would be greatly appreciated. :)

Last edited by a moderator: May 4, 2017