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Calculus/Volume of rotation around a slant

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Expert: Alon Mandes - 11/5/2008


Question

Graph
I know how to find the volume of rotation of a bounded area if it is rotated around the x or y-axis, but I have to do a problem where an area is rotated around a slant.

-Here is the description of the graph:

Let C be the arc of the curve y=f(x) between the points P(p,f(p)) and Q(q,f(q)) and let R be the region bounded by C, by the line y=mx+b (which lies entirely below C), and by the perpendiculars to the line from P and Q.

- First I have to show that the area R is:

(1/(1+m^2) X  integral( [f(x)-mx-b][1+mf'(x)], x, p, q)

-I also have to find the volume of rotation of the area around the slant to create an equation that is similar to the area equation.

I think I'm supposed to modify the coordinate system, but I'm not sure how to do that.  I also tried creating an equation for the sum of the rectangles within the area that I could develop into an integral, but I couldn't get anywhere with it.


Answer
Hi Libbie, I sent the question to the question pool, but I see
that no 1 solved it, so I decided to do so. I apologize for keeping
you waiting that long.

1st of all we know that the volume of rotation is :
V=π∫y(x)^2dx. Now the rotation is around another line. So we will
find transformation (X,Y) instead of (x,y) Where X=g(x) & calculate :
V=π∫Y(x(X))^2dx. So, the 1st step will be rotation of the axis in
an angle of α which tang(α)=m , & then we move the (0,0) to (0,-b/m).
A quick calculation of this transformation shows that :
X=(x+b/m)√(1+m^2) -> x=X/√(1+m^2) -b/m . This X is our new X-axis.
Now we need to find Y(X) :
According to Phytaguras : X^2+Y^2=x^2+f(x)^2. So,
Y(x)=sqrt[ f(x)^2+x^2-X^2 ]. Now we put X/√(1+m^2) -b/m instead of x.
Thus, Y(x)=sqrt[ f(X/√(1+m^2))^2 + (X/√(1+m^2) -b/m)^2 - X^2 ]
& the volume will become :
V=π∫Y(x(X))^2dx=∫[ f(X/√(1+m^2))^2 + (X/√(1+m^2) -b/m)^2 - X^2 ]dx.

Alon.

Alon Mandes

Expertise

Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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