Thread: Calc volume prob with integrals

Let R be the region bounded above by the curve y=x^2 , below by the line y=1 , and bounded on the right by the line x=2
Use the shell method to find the volume of the solid generated by revolving R around the line y=-3.


So what I have so far is the picture, and I know that the solid that the lines form looks somewhat like a dog bowl and is in the "cylindrical shells" category, but I am nt sure what formula to use for volume.
Since the generic formula for cylin. shells is V = integral(2pi * x * f(x) )dx, I'm not sure where this one fits in since the height of the cylinder is 4 and not just x like in the formula.

Anyone care to slave this one along with me???

Originally Posted by alex_nicole

Let R be the region bounded above by the curve y=x^2 , below by the line y=1 , and bounded on the right by the line x=2
Use the shell method to find the volume of the solid generated by revolving R around the line y=-3.


So what I have so far is the picture, and I know that the solid that the lines form looks somewhat like a dog bowl and is in the "cylindrical shells" category, but I am nt sure what formula to use for volume.
Since the generic formula for cylin. shells is V = integral(2pi * x * f(x) )dx, I'm not sure where this one fits in since the height of the cylinder is 4 and not just x like in the formula.

Anyone care to slave this one along with me???

The "x" in the formula there is actually the radius of each shell, whatever that happens to be. In your case, that distance will be represented by x + 3, since you are rotating around y = -3. The height of the cylinder, which is f(x), is going to depend on your x-value.

will the f(x) be the x^2 equation since that forms the outer curve???

Originally Posted by alex_nicole

will the f(x) be the x^2 equation since that forms the outer curve???

For this problem, since you are taking the region bounded above by and below by . Another question that needs answering is: what are the x-values between which you are going to integrate?

would the x-values be the region that the rotated bowl forms? How do i find those??? and were you saying that the f(x) equation IS the y=x^2 or the y=1 ?