How Is Volume Calculated by Integrating Around an Axis?

[CrazyAmerican]

Homework Statement

How do you find the volume of a region using integration?
the region bounded by
y = x^2
y = 0
x = 1
x = 4

and rotated about x = -1

Homework Equations

The Attempt at a Solution

it doesn't seem to me you can just use the outer radius squared minus the inner radius squared b/c if you draw it out, the graph is bound by x = 1, so you get this shoe looking thing instead of the graph continuing to the origin, which then you could do.

the outer radius would be 1+4 = 5
the inner radus would be 1+sqrt(y)

the limits of integration should be 0 to 16

 

[tiny-tim]

Hi CrazyAmerican!

it doesn't seem to me you can just use the outer radius squared minus the inner radius squared b/c if you draw it out, the graph is bound by x = 1, so you get this shoe looking thing instead of the graph continuing to the origin, which then you could do.

the outer radius would be 1+4 = 5
the inner radus would be 1+sqrt(y)

the limits of integration should be 0 to 16

(except of course that the inner radius is 1 for 0 ≤ y ≤ 1 )

I don't understand the difficulty … shoe? origin?

 

[HallsofIvy]

Every cross section of the solid, parallel to the x-axis, is a circle with center (-1, y) and radius the distance from (-1,y) to (x, y) on the curve. Since y= x^2, x= sqrt(y) (we know this is the positive root because x is between 1 and 4) so the radius is sqrt(y)-(-1)= sqrt(y)+ 1. Since this starts at x= 1, the "inner radius" is 1+1= 2 (not 1+ sqrt(y) nor 1) and the outer radius is 1+ sqrt(2). When x= 1, y= 1, when x= 4, y= 16 so the integration should be done from y= 1 to y= 16.