Solid of Revolution problem

[JustinJustinn]

(Please refer to the picture)

This problem is an exercise problem in our calculus class and we were tasked to find the volume using the Washer method. I already sketched it and know what the figure will look like if revolved: like a coin on its side with a funnel-shaped hole through it. But the problem is I don't know what is R (outer radius) and r (inner raidus) are. Is it R=x^2-2x+y=3, r=4-x?

 

[tkhunny]

First and foremost, don't get hung up with x's and y's. Use whatever is convenient and most obvious.
In this case, it's pretty obvious that:
R = 4
r = 4 - y
Worry about transformations later. If you need it in terms of x, then solve x^2-2x+y=3 for y = f(x) and you are on your way. If you need it in terms of y, then solve x^2-2x+y=3 for x = f(y) and you are on your way. (Pick the correct branch, of course - the one that contains (x=0,y=3)).

Second, and only slightly less important, since you are not under exam pressure, do it with shells, too. Keep trying both ways until you get the same answer both ways.

 

[JustinJustinn]

I have identified the outer and inner radius.
R=4
r= 4-(3+2x-x^2)= x^2-2x+1
And since it's only the second quadrant, it's from -1 to 0 only.
Is this correct?

 

[JustinJustinn]

And integrating pi ((4)^2-(x^2-2x+1)^2) from -1 to 0, I get 49pi/5 cubic units.

 

[tkhunny]

Do we get to see the Shells version or are you tired of this one?