## anonymous one year ago Find the volume of the solid formed by rotating the region bounded by the graph of y equals 1 plus the square root of x, the y-axis, and the line y = 3 about the y-axis.

1. anonymous
2. zepdrix

Ooo solids o revolution :) These are so fun

3. anonymous

eh, I'd rather play a game of tag

4. zepdrix

|dw:1444438841566:dw|Ok so you can see the region that we're dealing with.

5. zepdrix

|dw:1444439029610:dw|We're integrating from x=0 and to x= the intersection of these two curves.

6. zepdrix

$\large\rm y=3,\qquad\qquad y=1+\sqrt x$$\large\rm 3=1+\sqrt x$So what is our upper bound on x?

7. anonymous

4?

8. zepdrix

$\large\rm V=\int\limits_0^4 dv$Ok that takes care of that. Now let's cut a little slice into that region, and see if we can come up with an equation for the volumen of the shape that is spun around.

9. zepdrix

|dw:1444439321599:dw|We're going to cut a little slice, we'll say that it has "thickness" dx

10. zepdrix

|dw:1444439387037:dw|Looks like we get a cylindrical shell, ya? We have some information we need to figure out.

11. anonymous

So we need to find r and h?

12. zepdrix

Volume of a cylindrical shell is $$\large\rm v=(Circumference)(height)(Thickness)$$ Good, yes. We already know the thickness,$\large\rm dv=(Circumference)(height)(dx)$

13. zepdrix

|dw:1444439652366:dw|For this particular problem, r is the easy one, h is going to be a little more difficult to figure out.