PDA

View Full Version : help with calculus problem

mecbluefugate
10-29-2007, 07:12 AM
So I'm reviewing for an exam that I have tomorrow, and I came across these two similar problems. I realized I completely forgot how to do them. I know they both have to do with the integral of pi R squared, and something to do with outter radius - inner radius, but I am drawing a blank on how exactly to do them. Probably due to cramming, but somebody help me out with these please ---

1. Find the volume of the solid generated by rotating the region bounded by the curves y=xx and x=yy about the y-axis.

2. Find the volume of the solid generated by rotating the region bounded by the curves y=3x-xx and y=0 about the line x= -1.

Again, just trying to figure out how to do these problems. Since I'm reviewing for an exam, the process is much more important than the raw answer.

HallsofIvy
11-03-2007, 07:31 PM
So I'm reviewing for an exam that I have tomorrow, and I came across these two similar problems. I realized I completely forgot how to do them. I know they both have to do with the integral of pi R squared, and something to do with outter radius - inner radius, but I am drawing a blank on how exactly to do them. Probably due to cramming, but somebody help me out with these please ---
You don't have text book to consult? You will need to know more than "they have to do with"!

1. Find the volume of the solid generated by rotating the region bounded by the curves y=xx and x=yy about the y-axis.
There are two ways to do this. One method is to imagine a horizontal line segment between y= x^2 and x= y^2. Rotating that around the y-axis produces a "washer"- the area between the two circles. You can find the area be subtracting the area of the inner circle (y= x^2 give x= sqrt(y)) which is pi x^2= pi y from the area of the outer circle which is pi x^2= pi y^4: the area is pi(y^4- y). Imagining that washer to have thickness "dy" gives a volume pi(y^4- y)dy. Integrate that to find the volume of the whole thing.

The other method is to imagine a vertical line segment between y= x^2 and x= y^2. Rotating that around the y axis produces a thin cylinder. The area is the height of the cylinder, x^2- sqrt(x), times the circumference, pi x. Imagining that to have a thicknes dx, the volume of that thin cylinder is pi x(x^2- sqrt(x))dx. Integrate that to find the volume.

Do it both ways and see if you get the same answer!

2. Find the volume of the solid generated by rotating the region bounded by the curves y=3x-xx and y=0 about the line x= -1.
Same idea except instead of x values alone, you will need to use x-(-1)= x+1.